{"id":2666,"date":"2018-12-11T13:42:02","date_gmt":"2018-12-11T18:42:02","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-equations-using-determinants\/"},"modified":"2018-12-11T13:42:02","modified_gmt":"2018-12-11T18:42:02","slug":"solve-systems-of-equations-using-determinants","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/solve-systems-of-equations-using-determinants\/","title":{"raw":"Solve Systems of Equations Using Determinants","rendered":"Solve Systems of Equations Using Determinants"},"content":{"raw":"\n[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Evaluate the determinant of a \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) matrix<\/li><li>Evaluate the determinant of a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) matrix<\/li><li>Use Cramer\u2019s Rule to solve systems of equations<\/li><li>Solve applications using determinants<\/li><\/ul><\/div><div data-type=\"note\" class=\"be-prepared\"><p id=\"fs-id1167826995899\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167835531480\" type=\"1\"><li>Simplify: \\(5\\left(-2\\right)-\\left(-4\\right)\\left(1\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834556092\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(-3\\left(8-10\\right)+\\left(-2\\right)\\left(6-3\\right)-4\\left(-3-\\left(-4\\right)\\right).\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536158\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Simplify: \\(\\frac{-12}{-8}.\\)<div data-type=\"newline\"><br><\/div> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536325\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><p id=\"fs-id1167835421208\">In this section we will learn of another method to solve systems of linear equations called Cramer\u2019s rule. Before we can begin to use the rule, we need to learn some new definitions and notation.<\/p><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834517668\"><h3 data-type=\"title\">Evaluate the Determinant of a \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) Matrix<\/h3><p id=\"fs-id1167835287688\">If a matrix has the same number of rows and columns, we call it a <span data-type=\"term\">square matrix<\/span>. Each square matrix has a real number associated with it called its <span data-type=\"term\">determinant<\/span>. To find the determinant of the square matrix \\(\\left[\\begin{array}{c}a\\phantom{\\rule{1.5em}{0ex}}b\\hfill \\\\ c\\phantom{\\rule{1.5em}{0ex}}d\\hfill \\end{array}\\right],\\) we first write it as \\(|\\begin{array}{c}a\\phantom{\\rule{1.5em}{0ex}}b\\hfill \\\\ c\\phantom{\\rule{1.5em}{0ex}}d\\hfill \\end{array}|.\\) To get the real number value of the determinate we subtract the products of the diagonals, as shown.<\/p><span data-type=\"media\" data-alt=\"A 2 by 2 determinant is show, with its first row being a, b and second one being c, d. These values are written between two vertical lines instead of brackets as in the case of matrices. Two arrows are shown, one from a to d, the other from c to b. This determinant is equal to ad minus bc.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 2 determinant is show, with its first row being a, b and second one being c, d. These values are written between two vertical lines instead of brackets as in the case of matrices. Two arrows are shown, one from a to d, the other from c to b. This determinant is equal to ad minus bc.\"><\/span><div data-type=\"note\" id=\"fs-id1167835374978\"><div data-type=\"title\">Determinant<\/div><p id=\"fs-id1167834532264\">The determinant of any square matrix \\(\\left[\\begin{array}{c}a\\phantom{\\rule{1.5em}{0ex}}b\\hfill \\\\ c\\phantom{\\rule{1.5em}{0ex}}d\\hfill \\end{array}\\right],\\) where <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em> are real numbers, is<\/p><div data-type=\"equation\" id=\"fs-id1167835301633\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}a\\phantom{\\rule{1.5em}{0ex}}b\\hfill \\\\ c\\phantom{\\rule{1.5em}{0ex}}d\\hfill \\end{array}|=ad-bc\\)<\/div><\/div><div data-type=\"example\" id=\"fs-id1167835193012\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832058394\"><div data-type=\"problem\" id=\"fs-id1167835369363\"><p id=\"fs-id1167834432835\">Evaluate the determinate of <span class=\"token\">\u24d0<\/span> \\(\\left[\\begin{array}{c}4\\phantom{\\rule{1.5em}{0ex}}-2\\hfill \\\\ 3\\phantom{\\rule{1.5em}{0ex}}-1\\hfill \\end{array}\\right]\\) <span class=\"token\">\u24d1<\/span> \\(\\left[\\begin{array}{cccc}-3\\hfill &amp; &amp; &amp; \\hfill -4\\\\ -2\\hfill &amp; &amp; &amp; \\hfill 0\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832042643\"><p id=\"fs-id1167834191139\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167834462960\" class=\"unnumbered unstyled\" summary=\"Row 1 of the 2 by 2 matrix is 4, minus 2. Row 2 is 3, minus 1. The same is written in determinant form with diagonal arrows. Subtracting the products of the diagonals, we get 4 times minus 1 minus 3 times minus 2. We simplify to get 2.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"bottom\" data-align=\"left\">Write the determinant.<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832074051\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract the products of the diagonals.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830704243\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834309193\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831846606\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835282892\" class=\"unnumbered unstyled\" summary=\"Row 1 of the 2 by 2 matrix is minus 3, minus 4. Row 2 is minus 2, 0. The same is written in determinant form with diagonal arrows. Subtracting the products of the diagonals, we get minus 3 times 0 minus minus 2 times minus 4. We simplify to get minus 8.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"bottom\" data-align=\"left\">Write the determinant.<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066568\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Subtract the products of the diagonals.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826997005\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835226257\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835253835\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835280362\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834536041\"><div data-type=\"problem\" id=\"fs-id1167835422303\"><p id=\"fs-id1167832057952\">Evaluate the determinate of <span class=\"token\">\u24d0<\/span> \\(\\left[\\begin{array}{c}5\\phantom{\\rule{1.5em}{0ex}}-3\\hfill \\\\ 2\\phantom{\\rule{1.5em}{0ex}}-4\\hfill \\end{array}\\right]\\) <span class=\"token\">\u24d1<\/span> \\(\\left[\\begin{array}{cccc}\\hfill -4&amp; &amp; &amp; \\hfill -6\\\\ \\hfill 0&amp; &amp; &amp; \\hfill 7\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835208370\"><p id=\"fs-id1167831912980\"><span class=\"token\">\u24d0<\/span>\\(-14;\\)<span class=\"token\">\u24d1<\/span>\\(-28\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832055377\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834373486\"><div data-type=\"problem\" id=\"fs-id1167827902584\"><p id=\"fs-id1167834357131\">Evaluate the determinate of <span class=\"token\">\u24d0<\/span> \\(\\left[\\begin{array}{c}-1\\phantom{\\rule{1.5em}{0ex}}3\\hfill \\\\ -2\\phantom{\\rule{1.5em}{0ex}}4\\hfill \\end{array}\\right]\\) <span class=\"token\">\u24d1<\/span> \\(\\left[\\begin{array}{cccc}-7\\hfill &amp; &amp; &amp; \\hfill -3\\\\ -5\\hfill &amp; &amp; &amp; \\hfill 0\\end{array}\\right].\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826938221\"><p id=\"fs-id1167834423524\"><span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> \\(-15\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835350146\"><h3 data-type=\"title\">Evaluate the Determinant of a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) Matrix<\/h3><p id=\"fs-id1167835498954\">To evaluate the determinant of a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) matrix, we have to be able to evaluate the <span data-type=\"term\">minor of an entry<\/span> in the determinant. The minor of an entry is the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant found by eliminating the row and column in the \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant that contains the entry.<\/p><div data-type=\"note\" id=\"fs-id1167834124754\"><div data-type=\"title\">Minor of an entry in \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) a Determinant<\/div><p id=\"fs-id1167832065959\">The <strong data-effect=\"bold\">minor of an entry<\/strong> in a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant is the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant found by eliminating the row and column in the \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant that contains the entry.<\/p><\/div><p id=\"fs-id1167835338688\">To find the minor of entry \\({a}_{1},\\) we eliminate the row and column which contain it. So we eliminate the first row and first column. Then we write the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant that remains.<\/p><span data-type=\"media\" id=\"fs-id1167834423592\" data-alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. a1 is highlighted. Lines strike out the first row and the first column. What remains is called minor of a1. It is shown as a separate determinant whose first row is b2, c2 and second row is b3, c3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. a1 is highlighted. Lines strike out the first row and the first column. What remains is called minor of a1. It is shown as a separate determinant whose first row is b2, c2 and second row is b3, c3.\"><\/span><p id=\"fs-id1167835306351\">To find the minor of entry \\({b}_{2},\\) we eliminate the row and column that contain it. So we eliminate the 2<sup>nd<\/sup> row and 2<sup>nd<\/sup> column. Then we write the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant that remains.<\/p><span data-type=\"media\" id=\"fs-id1167832124422\" data-alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. b2 is highlighted. Lines strike out the second row and second column. What remains is minor of b2. It is written as a separate determinant whose first row is a1, c1 and second row is a3, c3.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. b2 is highlighted. Lines strike out the second row and second column. What remains is minor of b2. It is written as a separate determinant whose first row is a1, c1 and second row is a3, c3.\"><\/span><div data-type=\"example\" id=\"fs-id1167832052293\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167831887447\"><div data-type=\"problem\" id=\"fs-id1167830769653\"><p>For the determinant \\(|\\begin{array}{ccccccc}\\hfill 4&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 3\\\\ \\hfill 1&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\\\ \\hfill -2&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 2\\end{array}|,\\) find and then evaluate the minor of <span class=\"token\">\u24d0<\/span> \\({a}_{1}\\) <span class=\"token\">\u24d1<\/span> \\({b}_{3}\\) <span class=\"token\">\u24d2<\/span> \\({c}_{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834238985\"><p id=\"fs-id1167830954156\"><span class=\"token\">\u24d0<\/span><\/p><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835166304\" class=\"unnumbered unstyled\" summary=\"The first row of the 3 by 3 determinant is 4, minus 2, 3. Row 2 is 1, 0, minus 3. Row 3 is minus 2, minus 4, 2. Eliminating the row and column containing a1, we get the minor of a1. This 2 by 2 determinant has row 1: 0, minus 3 and row 2: minus 4, 2. Evaluate and simplify to get minus 12.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834502674\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"bottom\" data-align=\"left\">Eliminate the row and column that contains \\({a}_{1}.\\)<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254642\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant that remains.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831890624\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834048843\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834099451\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d1<\/span><div data-type=\"newline\"><br><\/div><table class=\"unnumbered unstyled\" summary=\"The 3 by 3 matrix has row 1: 4, minus 2, 3, row 2: 1, 0, minus 3 and row 3: minus 2, minus 4, 2. Eliminating the row and column containing b3, we get minor of b3 with row 1: 4, 3 and row 2: 1, minus 3. Evaluate and simplify to get minus 15.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"bottom\" data-align=\"left\">Eliminate the row and column that contains \\({b}_{3}.\\)<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831893602\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant that remains.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835342689\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834474237\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><p><\/p><div data-type=\"newline\"><br><\/div><div data-type=\"newline\"><br><\/div><span class=\"token\">\u24d2<\/span><div data-type=\"newline\"><br><\/div><table id=\"fs-id1167835303369\" class=\"unnumbered unstyled can-break\" summary=\"The 3 by 3 matrix has row 1: 4, minus 2, 3, row 2: 1, 0, minus 3 and row 3: minus 2, minus 4, 2. Eliminating the row and column containing c2, we get minor of c2 with row 1: 4, minus 2 and row 2: minus 2, 4. Evaluate and simplify to get 12\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831883033\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"bottom\" data-align=\"left\">Eliminate the row and column that contains \\({c}_{2}.\\)<\/td><td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832058516\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant that remains.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835348052\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826778954\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167830865964\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834423084\"><div data-type=\"problem\" id=\"fs-id1167834098196\"><p id=\"fs-id1167835310588\">For the determinant \\(|\\begin{array}{ccccccc}\\hfill 1&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 4\\\\ \\hfill 0&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -1\\\\ \\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 3\\end{array}|,\\) find and then evaluate the minor of <span class=\"token\">\u24d0<\/span> \\({a}_{1}\\) <span class=\"token\">\u24d1<\/span> \\({b}_{2}\\) <span class=\"token\">\u24d2<\/span> \\({c}_{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834429082\"><p><span class=\"token\">\u24d0<\/span> 3 <span class=\"token\">\u24d1<\/span> 11 <span class=\"token\">\u24d2<\/span> 2<\/p><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831037153\"><div data-type=\"problem\" id=\"fs-id1167835367305\"><p id=\"fs-id1167834238946\">For the determinant \\(|\\begin{array}{ccccccc}\\hfill -2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 0\\\\ \\hfill 3&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -1\\\\ \\hfill -1&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 3\\end{array}|,\\) find and then evaluate the minor of <span class=\"token\">\u24d0<\/span> \\({a}_{2}\\) <span class=\"token\">\u24d1<\/span> \\({b}_{3}\\) <span class=\"token\">\u24d2<\/span> \\({c}_{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834532254\"><p id=\"fs-id1167834340084\"><span class=\"token\">\u24d0<\/span>\\(-3\\)<span class=\"token\">\u24d1<\/span> 2 <span class=\"token\">\u24d2<\/span> 3<\/p><\/div><\/div><\/div><p id=\"fs-id1167835352856\">We are now ready to evaluate a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant. To do this we expand by minors, which allows us to evaluate the \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant using \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinants\u2014which we already know how to evaluate!<\/p><p id=\"fs-id1167835326066\">To evaluate a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant by expanding by minors along the first row, we use the following pattern:<\/p><span data-type=\"media\" id=\"fs-id1167834449187\" data-alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><\/span><p id=\"fs-id1167835422374\">Remember, to find the minor of an entry we eliminate the row and column that contains the entry.<\/p><div data-type=\"note\" id=\"fs-id1167834397786\"><div data-type=\"title\">Expanding by Minors along the First Row to Evaluate a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) Determinant<\/div><p id=\"fs-id1167830865570\">To evaluate a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant by <strong data-effect=\"bold\">expanding by minors along the first row<\/strong>, the following pattern:<\/p><span data-type=\"media\" id=\"fs-id1167835234815\" data-alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_021_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><\/span><\/div><div data-type=\"example\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835231574\"><div data-type=\"problem\" id=\"fs-id1167834179990\"><p id=\"fs-id1167835336341\">Evaluate the determinant \\(|\\begin{array}{ccccccc}\\hfill 2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -1\\\\ \\hfill 3&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 0\\\\ \\hfill -1&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -2\\end{array}|\\) by expanding by minors along the first row.<\/p><\/div><div data-type=\"solution\"><table id=\"fs-id1167826808290\" class=\"unnumbered unstyled\" summary=\"The first row of the determinant is 2, minus 3, minus 1. Row 2 is 3, 2, 0. Row 3 is minus 1, minus 1, minus 2. Expanding by minors, we get 2 times minor of 2 minus 3 times minor of 3 plus minus 1 times minor of minus 1. Evaluating each determinant and simplifying, we get minus 25.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835336567\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Expand by minors along the first row<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835369420\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate each determinant.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834196236\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835305817\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830696631\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835328645\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835418958\"><div data-type=\"problem\" id=\"fs-id1167834523938\"><p id=\"fs-id1167834376483\">Evaluate the determinant \\(|\\begin{array}{ccccccc}\\hfill 3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 4\\\\ \\hfill 0&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -2\\\\ \\hfill 2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill -1\\end{array}|,\\) by expanding by minors along the first row.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831025428\"><p id=\"fs-id1167835300734\">37<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834387531\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835354006\"><div data-type=\"problem\" id=\"fs-id1167835214250\"><p id=\"fs-id1167835530640\">Evaluate the determinant \\(|\\begin{array}{ccccccc}\\hfill 3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -2\\\\ \\hfill 2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 4\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\end{array}|,\\) by expanding by minors along the first row.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834438855\"><p id=\"fs-id1167835361021\">7<\/p><\/div><\/div><\/div><p>To evaluate a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.<\/p><p id=\"fs-id1167830697049\">When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.<\/p><div data-type=\"equation\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}+\\phantom{\\rule{1.5em}{0ex}}-\\phantom{\\rule{1.5em}{0ex}}+\\hfill \\\\ -\\phantom{\\rule{1.5em}{0ex}}+\\phantom{\\rule{1.5em}{0ex}}-\\hfill \\\\ +\\phantom{\\rule{1.5em}{0ex}}-\\phantom{\\rule{1.5em}{0ex}}+\\hfill \\end{array}|\\)<\/div><div data-type=\"note\" id=\"fs-id1167834383296\"><div data-type=\"title\">Sign Pattern<\/div><p id=\"fs-id1167830698084\">When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.<\/p><div data-type=\"equation\" id=\"fs-id1167835310672\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}+\\phantom{\\rule{1.5em}{0ex}}-\\phantom{\\rule{1.5em}{0ex}}+\\hfill \\\\ -\\phantom{\\rule{1.5em}{0ex}}+\\phantom{\\rule{1.5em}{0ex}}-\\hfill \\\\ +\\phantom{\\rule{1.5em}{0ex}}-\\phantom{\\rule{1.5em}{0ex}}+\\hfill \\end{array}|\\)<\/div><\/div><p id=\"fs-id1167834429687\">Notice that the sign pattern in the first row matches the signs between the terms in the expansion by the first row.<\/p><span data-type=\"media\" id=\"fs-id1167834555240\" data-alt=\"A 3 by 3 determinant has row 1: plus, minus, plus, row 2: minus, plus, minus and row 3: plus, minus, plus. The three signs in the first row each point to a minor determinant in the expansion of a 3 by 3 determinant. Plus points to minor of a1, minus to the minor of b1 and plus to the minor of c1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant has row 1: plus, minus, plus, row 2: minus, plus, minus and row 3: plus, minus, plus. The three signs in the first row each point to a minor determinant in the expansion of a 3 by 3 determinant. Plus points to minor of a1, minus to the minor of b1 and plus to the minor of c1.\"><\/span><p id=\"fs-id1167835367678\">Since we can expand by any row or column, how do we decide which row or column to use? Usually we try to pick a row or column that will make our calculation easier. If the determinant contains a 0, using the row or column that contains the 0 will make the calculations easier.<\/p><div data-type=\"example\" id=\"fs-id1167834186369\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835533969\"><div data-type=\"problem\" id=\"fs-id1167834257993\"><p id=\"fs-id1167835350318\">Evaluate the determinant \\(|\\begin{array}{ccccccc}4\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -3\\\\ 3\\hfill &amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill 2\\\\ 5\\hfill &amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill -3\\end{array}|\\) by expanding by minors.<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167835331737\">To expand by minors, we look for a row or column that will make our calculations easier. Since 0 is in the second row and second column, expanding by either of those is a good choice. Since the second row has fewer negatives than the second column, we will expand by the second row.<\/p><table id=\"fs-id1167832056052\" class=\"unnumbered unstyled\" summary=\"A 3 by 3 determinant has row 1 4, minus 1, minus 3, row 2: 3, 0, 2 and row 3 5, minus 4, minus 3. The second row is highlighted. Expand using the second row. Be careful of the signs. The middle row is minus, plus, minus. Expanding, we get minus 3 times minor of 3 plus 0 times minor of 0 minus 2 times minor of 2. Evaluating each minor determinant and simplifying, we get 49.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834524204\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Expand using the second row.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Be careful of the signs.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835326516\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate each determinant.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835511115\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835337580\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Add.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835420254\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826784031\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834429462\"><div data-type=\"problem\"><p id=\"fs-id1167835284588\">Evaluate the determinant \\(|\\begin{array}{ccccccc}2\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -3\\\\ 0\\hfill &amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill -4\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill -3\\end{array}|\\) by expanding by minors.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835213837\"><p id=\"fs-id1167826849510\">\\(-11\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835366417\" class=\"try\"><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835216554\"><p>Evaluate the determinant \\(|\\begin{array}{ccccccc}\\hfill -2&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -3\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 2\\\\ \\hfill 4&amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill 0\\end{array}|\\) by expanding by minors.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831833355\"><p id=\"fs-id1167826799095\">8<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834431184\"><h3 data-type=\"title\">Use Cramer\u2019s Rule to Solve Systems of Equations<\/h3><p id=\"fs-id1167835173740\">Cramer\u2019s Rule is a method of solving systems of equations using determinants. It can be derived by solving the general form of the systems of equations by elimination. Here we will demonstrate the rule for both systems of two equations with two variables and for systems of three equations with three variables.<\/p><p id=\"fs-id1167835328055\">Let\u2019s start with the systems of two equations with two variables.<\/p><div data-type=\"note\" id=\"fs-id1167831180556\"><div data-type=\"title\">Cramer\u2019s Rule for Solving a System of Two Equations<\/div><p>For the system of equations \\(\\left\\{\\begin{array}{c}{a}_{1}x+{b}_{1}y={k}_{1}\\hfill \\\\ {a}_{2}x+{b}_{2}y={k}_{2}\\hfill \\end{array},\\) the solution \\(\\left(x,y\\right)\\) can be determined by<\/p><span data-type=\"media\" data-alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants\"><\/span><\/div><p id=\"fs-id1167834099327\">Notice that to form the determinant <em data-effect=\"italics\">D<\/em>, we use take the coefficients of the variables.<\/p><span data-type=\"media\" id=\"fs-id1167830838415\" data-alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is D with row 1: a1, b1 and row 2: a2, b2. Column 1 has coefficients of x and column 2 has coefficients of\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is D with row 1: a1, b1 and row 2: a2, b2. Column 1 has coefficients of x and column 2 has coefficients of\"><\/span><p>Notice that to form the determinant \\({D}_{x}\\) and \\({D}_{y},\\) we substitute the constants for the coefficients of the variable we are finding.<\/p><span data-type=\"media\" data-alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is Dx has row 1: k1, b1 and row 2: k2, b2. Here columns 1 and 2 re constants and coefficients of y respectively. Determinant Dy has row 1: a1, k1 and row 2: a2, k2. Here, columns 1 and 2 are coefficients of x and constants respectively.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is Dx has row 1: k1, b1 and row 2: k2, b2. Here columns 1 and 2 re constants and coefficients of y respectively. Determinant Dy has row 1: a1, k1 and row 2: a2, k2. Here, columns 1 and 2 are coefficients of x and constants respectively.\"><\/span><div data-type=\"example\" id=\"fs-id1167834137622\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Solve a System of Equations Using Cramer\u2019s Rule<\/div><div data-type=\"exercise\" id=\"fs-id1167835225995\"><div data-type=\"problem\" id=\"fs-id1167826782724\"><p id=\"fs-id1167834526137\">Solve using Cramer\u2019s Rule: \\(\\left\\{\\begin{array}{c}2x+y=-4\\hfill \\\\ 3x-2y=-6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835629178\"><span data-type=\"media\" id=\"fs-id1167826782312\" data-alt=\"The equations are 2x plus y equals minus 4 and 3x minus 2y equals minus 6. Step 1. Evaluate the determinant D, using the coefficients of the variables. Determinant D has row 1: 2, 1 and row 2: 3, minus 2. So, D is minus 7.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 2x plus y equals minus 4 and 3x minus 2y equals minus 6. Step 1. Evaluate the determinant D, using the coefficients of the variables. Determinant D has row 1: 2, 1 and row 2: 3, minus 2. So, D is minus 7.\"><\/span><span data-type=\"media\" id=\"fs-id1167835215728\" data-alt=\"Step 2. Evaluate the determinant Dx. Use the constants in place of the x coefficients. We replace the coefficients of x, 2 and 3, with the constants, negative 4 and negative 6. We get Dx equal to 14.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Evaluate the determinant Dx. Use the constants in place of the x coefficients. We replace the coefficients of x, 2 and 3, with the constants, negative 4 and negative 6. We get Dx equal to 14.\"><\/span><span data-type=\"media\" id=\"fs-id1167835274917\" data-alt=\"Step 3. Evaluate the determinant Dy. Use the constants in place of the y coefficients. We replace the coefficients of y, 1 and 2, with the constants, negative 4 and negative 6. We get Dy equal to 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Evaluate the determinant Dy. Use the constants in place of the y coefficients. We replace the coefficients of y, 1 and 2, with the constants, negative 4 and negative 6. We get Dy equal to 0.\"><\/span><span data-type=\"media\" data-alt=\"Step 4. Find x and y. Substituting values of D, Dx and Dy in the equations x equal to Dx upon D and y equal to Dy upon D, we get x equal to minus 2 and y equal to 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Find x and y. Substituting values of D, Dx and Dy in the equations x equal to Dx upon D and y equal to Dy upon D, we get x equal to minus 2 and y equal to 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167835419109\" data-alt=\"Step 5. Write the solution as an ordered pair minus 2, 0.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5. Write the solution as an ordered pair minus 2, 0.\"><\/span><span data-type=\"media\" id=\"fs-id1167834396909\" data-alt=\"Step 6. Check that the ordered pair is a solution to both original equations.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6. Check that the ordered pair is a solution to both original equations.\"><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835389726\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831115647\"><div data-type=\"problem\" id=\"fs-id1167835267209\"><p id=\"fs-id1167835264828\">Solve using Cramer\u2019s rule: \\(\\left\\{\\begin{array}{c}3x+y=-3\\hfill \\\\ 2x+3y=6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831922888\"><p id=\"fs-id1167834279222\">\\(\\left(-\\frac{15}{7},\\frac{24}{7}\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834065903\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834161540\"><div data-type=\"problem\" id=\"fs-id1167835264028\"><p id=\"fs-id1167835511096\">Solve using Cramer\u2019s rule: \\(\\left\\{\\begin{array}{c}\\text{\u2212}x+y=2\\hfill \\\\ 2x+y=-4\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835423282\"><p id=\"fs-id1167834279427\">\\(\\left(-2,0\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834431139\" class=\"howto\"><div data-type=\"title\">Solve a system of two equations using Cramer\u2019s rule.<\/div><ol id=\"fs-id1167828401841\" type=\"1\" class=\"stepwise\"><li>Evaluate the determinant <em data-effect=\"italics\">D<\/em>, using the coefficients of the variables.<\/li><li>Evaluate the determinant \\({D}_{x}.\\) Use the constants in place of the <em data-effect=\"italics\">x<\/em> coefficients.<\/li><li>Evaluate the determinant \\({D}_{y}.\\) Use the constants in place of the <em data-effect=\"italics\">y<\/em> coefficients.<\/li><li>Find <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. \\(x=\\frac{{D}_{x}}{D},\\) \\(y=\\frac{{D}_{y}}{D}\\)<\/li><li>Write the solution as an ordered pair.<\/li><li>Check that the ordered pair is a solution to both original equations.<\/li><\/ol><\/div><p id=\"fs-id1167826778974\">To solve a system of three equations with three variables with Cramer\u2019s Rule, we basically do what we did for a system of two equations. However, we now have to solve for three variables to get the solution. The determinants are also going to be \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) which will make our work more interesting!<\/p><div data-type=\"note\"><div data-type=\"title\">Cramer\u2019s Rule for Solving a System of Three Equations<\/div><p id=\"fs-id1167835319418\">For the system of equations \\(\\left\\{\\begin{array}{c}{a}_{1}x+{b}_{1}y+{c}_{1}z={k}_{1}\\hfill \\\\ {a}_{2}x+{b}_{2}y+{c}_{2}z={k}_{2}\\hfill \\\\ {a}_{3}x+{b}_{3}y+{c}_{3}z={k}_{3}\\hfill \\end{array},\\) the solution \\(\\left(x,y,z\\right)\\) can be determined by<\/p><span data-type=\"media\" id=\"fs-id1167830865955\" data-alt=\"x is Dx upon D, y is Dy upon D and z is Dz upon D, where D is determinant with row 1: a1, b1, c1, row 2: a2, b2, c2, row 3: a3, b3, c3, use coefficients of the variables; Dx is determinant with row 1: k1, b1, c1, row 2: k2, b2, c2 and rwo 3: k3, b3, c3, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1, c1, row 2: a2, k2, c2 and row 3: a3, k3, c3, replace the y coefficients with constants; Dz is determinant with row 1: a1, b1, k1; row 2: a2, b2, k2, row 3: a3, b3, k3; replace the z coefficients with constants.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is Dx upon D, y is Dy upon D and z is Dz upon D, where D is determinant with row 1: a1, b1, c1, row 2: a2, b2, c2, row 3: a3, b3, c3, use coefficients of the variables; Dx is determinant with row 1: k1, b1, c1, row 2: k2, b2, c2 and rwo 3: k3, b3, c3, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1, c1, row 2: a2, k2, c2 and row 3: a3, k3, c3, replace the y coefficients with constants; Dz is determinant with row 1: a1, b1, k1; row 2: a2, b2, k2, row 3: a3, b3, k3; replace the z coefficients with constants.\"><\/span><\/div><div data-type=\"example\" id=\"fs-id1167835240630\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834279802\"><div data-type=\"problem\" id=\"fs-id1167835416019\"><p id=\"fs-id1167835345728\">Solve the system of equations using Cramer\u2019s Rule: \\(\\left\\{\\begin{array}{c}3x-5y+4z=5\\hfill \\\\ 5x+2y+z=0\\hfill \\\\ 2x+3y-2z=3\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835233369\"><table id=\"fs-id1167835311187\" class=\"unnumbered unstyled can-break\" summary=\"Evaluate the determinant D. D has row 1: 3, minus 5, 4. Row 2 is 5, 2, 1. Row 3 is 2, 3, minus 2. Expand by minors using column 1. Column 1 has the signs plus minus plus. D is 3 times first minor minus 5 times second minor plus 2 times third minor where the first minor has row 1: 2, 1 and row 2: 3, minus 2; second minor has row 1: minus 5, 4 and row 2: 3, minus 2; third minor has row 1: minus 5, 4 and row 2: 2, 1. Evaluate the determinants and simplify to get D equal to minus 37. To evaluate the determinant Dx, use the constants to replace the coefficients of x. Expand by minors using column 1. Evaluate and simplify to get Dx equal to minus 74. To evaluate the determinant Dy, use the constants to replace the coefficients of y. Expand by minors using column 2. Evaluate and simplify to get Dy equal to 111. To evaluate the determinant Dz, use the constants to replace the coefficients of z. Expand by minors using column 3. Evaluate and simplify to get Dz equal to 148. Find x, y, z and write the ordered triple 2, minus 3, minus 4. Check.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinant <em data-effect=\"italics\">D<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826801782\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Expand by minors using column 1.<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835504018\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835235110\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834279238\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832152001\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826938247\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834131323\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinant \\({D}_{x}.\\) Use the<div data-type=\"newline\"><br><\/div>constants to replace the coefficients of <em data-effect=\"italics\">x<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Expand by minors using column 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835634609\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831825107\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834556769\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834517442\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinant \\({D}_{y}.\\) Use the<div data-type=\"newline\"><br><\/div>constants to replace the coefficients of <em data-effect=\"italics\">y<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835376143\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834408455\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826798788\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835335956\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831959318\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327732\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835259317\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinant \\({D}_{z}.\\) Use the<div data-type=\"newline\"><br><\/div>constants to replace the coefficients of <em data-effect=\"italics\">z<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835423151\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834587839\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018v_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835367199\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018w_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834423610\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018x_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835325481\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018y_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237771\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018z_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find <em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>, and <em data-effect=\"italics\">z<\/em>.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830697629\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018aa_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018bb_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832052669\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018cc_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered triple.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835330774\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018dd_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"bottom\" data-align=\"left\">Check that the ordered triple is a solution<div data-type=\"newline\"><br><\/div>to <strong data-effect=\"bold\">all three<\/strong> original equations.<\/td><td data-valign=\"bottom\" data-align=\"left\">We leave the check to you.<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The solution is \\(\\left(2,-3,-4\\right).\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835167506\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835358592\"><div data-type=\"problem\" id=\"fs-id1167835192469\"><p id=\"fs-id1167835323641\">Solve the system of equations using Cramer\u2019s Rule: \\(\\left\\{\\begin{array}{c}3x+8y+2z=-5\\hfill \\\\ 2x+5y-3z=0\\hfill \\\\ x+2y-2z=-1\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835363688\"><p id=\"fs-id1167831893121\">\\(\\left(-9,3,-1\\right)\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834433665\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835308933\"><div data-type=\"problem\" id=\"fs-id1167826783812\"><p id=\"fs-id1167835421498\">Solve the system of equations using Cramer\u2019s Rule: \\(\\left\\{\\begin{array}{c}3x+y-6z=-3\\hfill \\\\ 2x+6y+3z=0\\hfill \\\\ 3x+2y-3z=-6\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834183850\"><p id=\"fs-id1167835165883\">\\(\\left(-6,3,-2\\right)\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835344810\">Cramer\u2019s rule does not work when the value of the <em data-effect=\"italics\">D<\/em> determinant is 0, as this would mean we would be dividing by 0. But when \\(D=0,\\) the system is either inconsistent or dependent.<\/p><p id=\"fs-id1167835310719\">When the value of \\(D=0\\) and \\({D}_{x},{D}_{y}\\) and \\({D}_{z}\\) are all zero, the system is consistent and dependent and there are infinitely many solutions.<\/p><p id=\"fs-id1167831040599\">When the value of \\(D=0\\) and \\({D}_{x},{D}_{y}\\) and \\({D}_{z}\\) are not all zero, the system is inconsistent and there is no solution.<\/p><div data-type=\"note\" id=\"fs-id1167835253988\"><div data-type=\"title\">Dependent and Inconsistent Systems of Equations<\/div><p id=\"fs-id1167835325469\">For any system of equations, where the <strong data-effect=\"bold\">value of the determinant<\/strong> \\(D=0,\\)<\/p><div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccccc}\\mathbf{\\text{Value of determinants}}\\hfill &amp; &amp; &amp; \\mathbf{\\text{Type of system}}\\hfill &amp; &amp; &amp; \\mathbf{\\text{Solution}}\\hfill \\\\ D=0\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{x},{D}_{y}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{z}\\phantom{\\rule{0.2em}{0ex}}\\text{are all zero}\\hfill &amp; &amp; &amp; \\text{consistent and dependent}\\hfill &amp; &amp; &amp; \\text{infinitely many solutions}\\hfill \\\\ D=0\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{x},{D}_{y}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{z}\\phantom{\\rule{0.2em}{0ex}}\\text{are not all zero}\\hfill &amp; &amp; &amp; \\text{inconsistent}\\hfill &amp; &amp; &amp; \\text{no solution}\\hfill \\end{array}\\)<\/div><p id=\"fs-id1168757812384\">In the next example, we will use the values of the determinants to find the solution of the system.<\/p><div data-type=\"example\" id=\"fs-id1167834537639\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167827943773\"><div data-type=\"problem\" id=\"fs-id1167834094605\"><p id=\"fs-id1167834246685\">Solve the system of equations using Cramer\u2019s rule : \\(\\left\\{\\begin{array}{c}\\hfill x+3y=4\\\\ \\hfill -2x-6y=3\\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835306172\"><p id=\"fs-id1167831923781\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; &amp; &amp; \\left\\{\\begin{array}{c}\\hfill x+3y=4\\\\ \\hfill \\text{\u2212}2x-6y=3\\end{array}\\hfill \\\\ \\begin{array}{c}\\text{Evaluate the determinant}\\phantom{\\rule{0.2em}{0ex}}D,\\phantom{\\rule{0.2em}{0ex}}\\text{using the}\\hfill \\\\ \\text{coefficients of the variables.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; D=|\\begin{array}{cccc}\\hfill 1&amp; &amp; &amp; \\hfill 3\\\\ \\hfill -2&amp; &amp; &amp; \\hfill -6\\end{array}|\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; D=-6-\\left(-6\\right)\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; D=0\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167831881762\">We cannot use Cramer\u2019s Rule to solve this system. But by looking at the value of the determinants \\({D}_{x}\\) and \\({D}_{y},\\) we can determine whether the system is dependent or inconsistent.<\/p><p id=\"fs-id1167832058741\">\\(\\begin{array}{cccccc}\\text{Evaluate the determinant}\\phantom{\\rule{0.2em}{0ex}}{D}_{x}.\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3.5em}{0ex}}{D}_{x}=|\\begin{array}{cccc}4\\hfill &amp; &amp; &amp; \\hfill 3\\\\ 3\\hfill &amp; &amp; &amp; \\hfill -6\\end{array}|\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3.5em}{0ex}}{D}_{x}=-24-9\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3.5em}{0ex}}{D}_{x}=15\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167834524879\">Since all the determinants are not zero, the system is inconsistent. There is no solution.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832075354\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831835395\"><div data-type=\"problem\" id=\"fs-id1167834555551\"><p id=\"fs-id1167835194834\">Solve the system of equations using Cramer\u2019s rule: \\(\\left\\{\\begin{array}{c}4x-3y=8\\hfill \\\\ 8x-6y=14\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826941225\"><p id=\"fs-id1167834537917\">no solution<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835339843\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835362814\"><div data-type=\"problem\" id=\"fs-id1167835363740\"><p id=\"fs-id1167835310829\">Solve the system of equations using Cramer\u2019s rule: \\(\\left\\{\\begin{array}{c}x=-3y+4\\hfill \\\\ 2x+6y=8\\hfill \\end{array}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835352006\"><p id=\"fs-id1167835207369\">infinite solutions<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831092182\"><h3 data-type=\"title\">Solve Applications using Determinants<\/h3><p id=\"fs-id1167835417818\">An interesting application of determinants allows us to test if points are collinear. Three points \\(\\left({x}_{1},{y}_{1}\\right),\\) \\(\\left({x}_{2},{y}_{2}\\right)\\) and \\(\\left({x}_{3},{y}_{3}\\right)\\) are collinear if and only if the determinant below is zero.<\/p><div data-type=\"equation\" id=\"fs-id1167835623297\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}{x}_{1}\\phantom{\\rule{1.5em}{0ex}}{y}_{1}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\\\ {x}_{2}\\phantom{\\rule{1.5em}{0ex}}{y}_{2}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\\\ {x}_{3}\\phantom{\\rule{1.5em}{0ex}}{y}_{3}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\end{array}|=0\\)<\/div><div data-type=\"note\" id=\"fs-id1167835345043\"><div data-type=\"title\">Test for Collinear Points<\/div><p id=\"fs-id1167835190915\">Three points \\(\\left({x}_{1},{y}_{1}\\right),\\) \\(\\left({x}_{2},{y}_{2}\\right)\\) and \\(\\left({x}_{3},{y}_{3}\\right)\\) are collinear if and only if<\/p><div data-type=\"equation\" id=\"fs-id1167835575471\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}{x}_{1}\\phantom{\\rule{1.5em}{0ex}}{y}_{1}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\\\ {x}_{2}\\phantom{\\rule{1.5em}{0ex}}{y}_{2}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\\\ {x}_{3}\\phantom{\\rule{1.5em}{0ex}}{y}_{3}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\end{array}|=0\\)<\/div><\/div><p id=\"fs-id1167832075915\">We will use this property in the next example.<\/p><div data-type=\"example\" id=\"fs-id1167835337787\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835468588\"><div data-type=\"problem\" id=\"fs-id1167835230111\"><p id=\"fs-id1167831824215\">Determine whether the points \\(\\left(5,-5\\right),\\) \\(\\left(4,-3\\right),\\) and \\(\\left(3,-1\\right)\\) are collinear.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832054072\"><table id=\"fs-id1167834494992\" class=\"unnumbered unstyled\" summary=\"The 3 by 3 determinant has the last column with all ones. Substitute the values into the determinant. We get row 1: 5, minus 5, 1, row 2: 4, minus 3, 1 and row 3: 3, minus 1, 1. Evaluate the determinant by expanding by minors using column 3. Evaluate the determinants. Simplify to get D equal to 0. The value of the determinate is 0, so the points are collinear.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834161471\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Substitute the values into the determinant.<div data-type=\"newline\"><br><\/div>\\(\\left(5,-5\\right),\\) \\(\\left(4,-3\\right),\\) and \\(\\left(3,-1\\right)\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830704055\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinant by expanding<div data-type=\"newline\"><br><\/div>by minors using column 3.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832052912\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835363686\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834382391\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835510774\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\"><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\">The value of the determinate is 0, so the<div data-type=\"newline\"><br><\/div>points are collinear.<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826871537\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831023875\"><div data-type=\"problem\" id=\"fs-id1167831066155\"><p id=\"fs-id1167832052184\">Determine whether the points \\(\\left(3,-2\\right),\\) \\(\\left(5,-3\\right),\\) and \\(\\left(1,-1\\right)\\) are collinear.<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167828326434\">yes<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832066110\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835304607\"><div data-type=\"problem\" id=\"fs-id1167835346877\"><p id=\"fs-id1167835329140\">Determine whether the points \\(\\left(-4,-1\\right),\\) \\(\\left(-6,2\\right),\\) and \\(\\left(-2,-4\\right)\\) are collinear.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834064114\"><p id=\"fs-id1167831836301\">yes<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834196491\" class=\"media-2\"><p id=\"fs-id1167834244267\">Access these online resources for additional instruction and practice with solving systems of linear inequalities by graphing.<\/p><ul id=\"fs-id1167830963112\" data-display=\"block\"><li><a href=\"https:\/\/www.openstax.org\/l\/37syslinineqgph\">Solving Systems of Linear Inequalities by Graphing<\/a><\/li><li><a href=\"https:\/\/www.openstax.org\/l\/37syslineqs\">Systems of Linear Inequalities<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831833252\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835300769\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Determinant:<\/strong> The determinant of any square matrix \\(\\left[\\begin{array}{c}a\\phantom{\\rule{1.5em}{0ex}}b\\hfill \\\\ c\\phantom{\\rule{1.5em}{0ex}}d\\hfill \\end{array}\\right],\\) where <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em> are real numbers, is<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167830704326\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}a\\phantom{\\rule{1.5em}{0ex}}b\\hfill \\\\ c\\phantom{\\rule{1.5em}{0ex}}d\\hfill \\end{array}|=ad-bc\\)<\/div><\/li><li><strong data-effect=\"bold\">Expanding by Minors along the First Row to Evaluate a 3 \u00d7 3 Determinant:<\/strong> To evaluate a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant by expanding by minors along the first row, the following pattern:<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167835303268\" data-alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><\/span> <\/li><li><strong data-effect=\"bold\">Sign Pattern:<\/strong> When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167834132743\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}+\\phantom{\\rule{1.5em}{0ex}}-\\phantom{\\rule{1.5em}{0ex}}+\\hfill \\\\ -\\phantom{\\rule{1.5em}{0ex}}+\\phantom{\\rule{1.5em}{0ex}}-\\hfill \\\\ +\\phantom{\\rule{1.5em}{0ex}}-\\phantom{\\rule{1.5em}{0ex}}+\\hfill \\end{array}|\\)<\/div><\/li><li><strong data-effect=\"bold\">Cramer\u2019s Rule:<\/strong> For the system of equations \\(\\left\\{\\begin{array}{c}{a}_{1}x+{b}_{1}y={k}_{1}\\hfill \\\\ {a}_{2}x+{b}_{2}y={k}_{2}\\hfill \\end{array},\\) the solution \\(\\left(x,y\\right)\\) can be determined by<div data-type=\"newline\"><br><\/div> <span data-type=\"media\" id=\"fs-id1167832015822\" data-alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants.\"><\/span><div data-type=\"newline\"><br><\/div> Notice that to form the determinant <em data-effect=\"italics\">D<\/em>, we use take the coefficients of the variables.<\/li><li><strong data-effect=\"bold\">How to solve a system of two equations using Cramer\u2019s rule.<\/strong><ol id=\"fs-id1167835239369\" type=\"1\" class=\"stepwise\"><li>Evaluate the determinant <em data-effect=\"italics\">D<\/em>, using the coefficients of the variables.<\/li><li>Evaluate the determinant \\({D}_{x}.\\) Use the constants in place of the <em data-effect=\"italics\">x<\/em> coefficients.<\/li><li>Evaluate the determinant \\({D}_{y}.\\) Use the constants in place of the <em data-effect=\"italics\">y<\/em> coefficients.<\/li><li>Find <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. \\(x=\\frac{{D}_{x}}{D},\\) \\(y=\\frac{{D}_{y}}{D}.\\)<\/li><li>Write the solution as an ordered pair.<\/li><li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li><li><strong data-effect=\"bold\">Dependent and Inconsistent Systems of Equations:<\/strong> For any system of equations, where the <strong data-effect=\"bold\">value of the determinant<\/strong> \\(D=0,\\)<div data-type=\"newline\"><br><\/div> \\(\\begin{array}{ccccccc}\\mathbf{\\text{Value of determinants}}\\hfill &amp; &amp; &amp; \\mathbf{\\text{Type of system}}\\hfill &amp; &amp; &amp; \\mathbf{\\text{Solution}}\\hfill \\\\ D=0\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{x},{D}_{y}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{z}\\phantom{\\rule{0.2em}{0ex}}\\text{are all zero}\\hfill &amp; &amp; &amp; \\text{consistent and dependent}\\hfill &amp; &amp; &amp; \\text{infinitely many solutions}\\hfill \\\\ D=0\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{x},{D}_{y}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{D}_{z}\\phantom{\\rule{0.2em}{0ex}}\\text{are not all zero}\\hfill &amp; &amp; &amp; \\text{inconsistent}\\hfill &amp; &amp; &amp; \\text{no solution}\\hfill \\end{array}\\)<\/li><li><strong data-effect=\"bold\">Test for Collinear Points:<\/strong> Three points \\(\\left({x}_{1},{y}_{1}\\right),\\) \\(\\left({x}_{2},{y}_{2}\\right),\\) and \\(\\left({x}_{3},{y}_{3}\\right)\\) are collinear if and only if<div data-type=\"newline\"><br><\/div> <div data-type=\"equation\" id=\"fs-id1167826997386\" class=\"unnumbered\" data-label=\"\">\\(|\\begin{array}{c}{x}_{1}\\phantom{\\rule{1.5em}{0ex}}{y}_{1}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\\\ {x}_{2}\\phantom{\\rule{1.5em}{0ex}}{y}_{2}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\\\ {x}_{3}\\phantom{\\rule{1.5em}{0ex}}{y}_{3}\\phantom{\\rule{1.5em}{0ex}}1\\hfill \\end{array}|=0\\)<\/div><\/li><\/ol><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835419241\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167832015675\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167834345575\"><strong data-effect=\"bold\">Evaluate the Determinant of a 2 \u00d7 2 Matrix<\/strong><\/p><p id=\"fs-id1167834190283\">In the following exercises, evaluate the determinate of each square matrix.<\/p><div data-type=\"exercise\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834403205\"><p id=\"fs-id1167834556820\">\\(\\left[\\begin{array}{c}6\\phantom{\\rule{1.5em}{0ex}}-2\\hfill \\\\ 3\\phantom{\\rule{1.5em}{0ex}}-1\\hfill \\end{array}\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835511346\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835333670\"><p id=\"fs-id1167835362153\">\\(\\left[\\begin{array}{c}-4\\phantom{\\rule{1.5em}{0ex}}8\\hfill \\\\ -3\\phantom{\\rule{1.5em}{0ex}}5\\hfill \\end{array}\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826778783\"><p id=\"fs-id1167835339455\">4<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834395430\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835378837\"><p id=\"fs-id1167835326831\">\\(\\left[\\begin{array}{cccc}\\hfill -3&amp; &amp; &amp; \\hfill 5\\\\ \\hfill 0&amp; &amp; &amp; \\hfill -4\\end{array}\\right]\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834395742\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834396530\"><p id=\"fs-id1167835281881\">\\(\\left[\\begin{array}{cccc}\\hfill -2&amp; &amp; &amp; \\hfill 0\\\\ \\hfill 7&amp; &amp; &amp; \\hfill -5\\end{array}\\right]\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835378046\"><p id=\"fs-id1167835353456\">10<\/p><\/div><\/div><p id=\"fs-id1167835321872\"><strong data-effect=\"bold\">Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/strong><\/p><p id=\"fs-id1167834505764\">In the following exercises, find and then evaluate the indicated minors.<\/p><div data-type=\"exercise\" id=\"fs-id1167835595008\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835609663\"><p id=\"fs-id1167835329320\">\\(|\\begin{array}{ccccccc}\\hfill 3&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 4\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -2\\\\ \\hfill -4&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill 5\\end{array}|\\)<\/p><div data-type=\"newline\"><br><\/div>Find the minor <span class=\"token\">\u24d0<\/span>\\({a}_{1}\\)<span class=\"token\">\u24d1<\/span>\\({b}_{2}\\)<span class=\"token\">\u24d2<\/span>\\({c}_{3}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835237643\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831833282\"><p id=\"fs-id1167831882030\">\\(|\\begin{array}{ccccccc}\\hfill -1&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 2\\\\ \\hfill 4&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -1\\\\ \\hfill -2&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\end{array}|\\)<\/p><div data-type=\"newline\"><br><\/div>Find the minor <span class=\"token\">\u24d0<\/span>\\({a}_{1}\\)<span class=\"token\">\u24d1<\/span>\\({b}_{1}\\)<span class=\"token\">\u24d2<\/span>\\({c}_{2}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167834182338\"><p id=\"fs-id1167835367765\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> \\(-14\\) <span class=\"token\">\u24d2<\/span> \\(-6\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834372737\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835283519\"><p id=\"fs-id1167835257388\">\\(|\\begin{array}{ccccccc}\\hfill 2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -4\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -3\\\\ \\hfill 0&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -2\\end{array}|\\)<\/p><div data-type=\"newline\"><br><\/div>Find the minor <span class=\"token\">\u24d0<\/span>\\({a}_{2}\\)<span class=\"token\">\u24d1<\/span>\\({b}_{2}\\)<span class=\"token\">\u24d2<\/span>\\({c}_{2}\\)<\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826994627\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834463841\"><p id=\"fs-id1167834512660\">\\(|\\begin{array}{ccccccc}\\hfill -2&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 3\\\\ \\hfill 1&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill 0\\\\ \\hfill -2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill -2\\end{array}|\\)<\/p><div data-type=\"newline\"><br><\/div>Find the minor <span class=\"token\">\u24d0<\/span>\\({a}_{3}\\)<span class=\"token\">\u24d1<\/span>\\({b}_{3}\\)<span class=\"token\">\u24d2<\/span>\\({c}_{3}\\)<\/div><div data-type=\"solution\" id=\"fs-id1167835509678\"><p id=\"fs-id1167835377182\"><span class=\"token\">\u24d0<\/span> 9 <span class=\"token\">\u24d1<\/span> \\(-3\\) <span class=\"token\">\u24d2<\/span> 8<\/p><\/div><\/div><p id=\"fs-id1167830865847\">In the following exercises, evaluate each determinant by expanding by minors along the first row.<\/p><div data-type=\"exercise\" id=\"fs-id1167834185930\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830961579\"><p id=\"fs-id1167835585208\">\\(|\\begin{array}{ccccccc}\\hfill -2&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill -1\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill -2\\\\ \\hfill 3&amp; &amp; &amp; \\hfill 1&amp; &amp; &amp; \\hfill -3\\end{array}|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835309557\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835233219\"><p id=\"fs-id1167832153428\">\\(|\\begin{array}{ccccccc}\\hfill 4&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -2\\\\ \\hfill -3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 1\\\\ \\hfill -2&amp; &amp; &amp; \\hfill -5&amp; &amp; &amp; \\hfill 7\\end{array}|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834061982\"><p id=\"fs-id1167832074771\">\\(-77\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835370962\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835309883\"><p id=\"fs-id1167835319884\">\\(|\\begin{array}{ccccccc}\\hfill -2&amp; &amp; &amp; \\hfill -3&amp; &amp; &amp; \\hfill -4\\\\ \\hfill 5&amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill 7\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 0\\end{array}|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835510649\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835186869\"><p id=\"fs-id1167832042307\">\\(|\\begin{array}{ccccccc}1\\hfill &amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill -2\\\\ 5\\hfill &amp; &amp; &amp; \\hfill -6&amp; &amp; &amp; \\hfill 4\\\\ 0\\hfill &amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill -1\\end{array}|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830962352\"><p id=\"fs-id1167831071407\">49<\/p><\/div><\/div><p id=\"fs-id1167831921120\">In the following exercises, evaluate each determinant by expanding by minors.<\/p><div data-type=\"exercise\" id=\"fs-id1167831919481\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834234224\"><p id=\"fs-id1167835496226\">\\(|\\begin{array}{ccccccc}\\hfill -5&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -4\\\\ \\hfill 4&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill -3\\\\ \\hfill 2&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 6\\end{array}|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834552380\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835417704\"><p id=\"fs-id1167835305417\">\\(|\\begin{array}{ccccccc}\\hfill 4&amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill 3\\\\ \\hfill 3&amp; &amp; &amp; \\hfill -2&amp; &amp; &amp; \\hfill 2\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 0&amp; &amp; &amp; \\hfill 4\\end{array}|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830699189\"><p id=\"fs-id1167834120481\">\\(-24\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835200448\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835200450\"><p id=\"fs-id1167834377196\">\\(|\\begin{array}{ccccccc}\\hfill 3&amp; &amp; &amp; \\hfill 5&amp; &amp; &amp; \\hfill 4\\\\ \\hfill -1&amp; &amp; &amp; \\hfill 3&amp; &amp; &amp; \\hfill 0\\\\ \\hfill -2&amp; &amp; &amp; \\hfill 6&amp; &amp; &amp; \\hfill 1\\end{array}|\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835331651\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834433398\"><p id=\"fs-id1167834433400\">\\(|\\begin{array}{ccccccc}2\\hfill &amp; &amp; &amp; \\hfill -4&amp; &amp; &amp; \\hfill -3\\\\ 5\\hfill &amp; &amp; &amp; \\hfill -1&amp; &amp; &amp; \\hfill -4\\\\ 3\\hfill &amp; &amp; &amp; \\hfill 2&amp; &amp; &amp; \\hfill 0\\end{array}|\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835416340\"><p>25<\/p><\/div><\/div><p id=\"fs-id1167835344963\"><strong data-effect=\"bold\">Use Cramer\u2019s Rule to Solve Systems of Equations<\/strong><\/p><p id=\"fs-id1167835301822\">In the following exercises, solve each system of equations using Cramer\u2019s Rule.<\/p><div data-type=\"exercise\" id=\"fs-id1167835340758\"><div data-type=\"problem\" id=\"fs-id1167835340760\"><p id=\"fs-id1167835367493\">\\(\\left\\{\\begin{array}{c}-2x+3y=3\\hfill \\\\ x+3y=12\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826783768\"><div data-type=\"problem\" id=\"fs-id1167832226433\"><p id=\"fs-id1167832226435\">\\(\\left\\{\\begin{array}{c}x-2y=-5\\hfill \\\\ 2x-3y=-4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826972439\"><p id=\"fs-id1167834090753\">\\(\\left(7,6\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835595911\"><div data-type=\"problem\" id=\"fs-id1167835595913\"><p id=\"fs-id1167831890391\">\\(\\left\\{\\begin{array}{c}x-3y=-9\\hfill \\\\ 2x+5y=4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835321848\"><div data-type=\"problem\" id=\"fs-id1167835341103\"><p id=\"fs-id1167835166217\">\\(\\left\\{\\begin{array}{c}2x+y=-4\\hfill \\\\ 3x-2y=-6\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834191310\"><p id=\"fs-id1167834191312\">\\(\\left(-2,0\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834064990\"><div data-type=\"problem\" id=\"fs-id1167834191302\"><p id=\"fs-id1167834191305\">\\(\\left\\{\\begin{array}{c}x-2y=-5\\hfill \\\\ 2x-3y=-4\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835364450\"><div data-type=\"problem\" id=\"fs-id1167832151802\"><p id=\"fs-id1167832151804\">\\(\\left\\{\\begin{array}{c}x-3y=-9\\hfill \\\\ 2x+5y=4\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832123911\"><p id=\"fs-id1167832056865\">\\(\\left(-3,2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831985621\"><div data-type=\"problem\" id=\"fs-id1167831985623\"><p id=\"fs-id1167835306116\">\\(\\left\\{\\begin{array}{c}5x-3y=-1\\hfill \\\\ 2x-y=2\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835333914\"><div data-type=\"problem\" id=\"fs-id1167835344196\"><p>\\(\\left\\{\\begin{array}{c}3x+8y=-3\\hfill \\\\ 2x+5y=-3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835166504\"><p id=\"fs-id1167835334618\">\\(\\left(-9,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826977931\"><div data-type=\"problem\"><p id=\"fs-id1167834536120\">\\(\\left\\{\\begin{array}{c}6x-5y+2z=3\\hfill \\\\ 2x+y-4z=5\\hfill \\\\ 3x-3y+z=-1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831911605\"><div data-type=\"problem\" id=\"fs-id1167831911607\"><p id=\"fs-id1167834133506\">\\(\\left\\{\\begin{array}{c}4x-3y+z=7\\hfill \\\\ 2x-5y-4z=3\\hfill \\\\ 3x-2y-2z=-7\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826874676\"><p id=\"fs-id1167832066827\">\\(\\left(-3,-5,4\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835381001\"><div data-type=\"problem\" id=\"fs-id1167834372362\"><p id=\"fs-id1167834372365\">\\(\\left\\{\\begin{array}{c}2x-5y+3z=8\\hfill \\\\ 3x-y+4z=7\\hfill \\\\ x+3y+2z=-3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826998295\"><div data-type=\"problem\" id=\"fs-id1167834222347\"><p id=\"fs-id1167834222349\">\\(\\left\\{\\begin{array}{c}11x+9y+2z=-9\\hfill \\\\ 7x+5y+3z=-7\\hfill \\\\ 4x+3y+z=-3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835333189\"><p id=\"fs-id1167835333191\">\\(\\left(2,-3,-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835280485\"><div data-type=\"problem\" id=\"fs-id1167835280487\"><p id=\"fs-id1167835328954\">\\(\\left\\{\\begin{array}{c}x+2z=0\\hfill \\\\ 4y+3z=-2\\hfill \\\\ 2x-5y=3\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\"><p id=\"fs-id1167835347394\">\\(\\left\\{\\begin{array}{c}2x+5y=4\\hfill \\\\ 3y-z=3\\hfill \\\\ 4x+3z=-3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831910832\"><p id=\"fs-id1167831910834\">\\(\\left(-3,2,3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826978776\"><div data-type=\"problem\" id=\"fs-id1167826978778\"><p id=\"fs-id1167834423452\">\\(\\left\\{\\begin{array}{c}2y+3z=-1\\hfill \\\\ 5x+3y=-6\\hfill \\\\ 7x+z=1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830700950\"><div data-type=\"problem\" id=\"fs-id1167830700953\"><p id=\"fs-id1167834537019\">\\(\\left\\{\\begin{array}{c}3x-z=-3\\hfill \\\\ 5y+2z=-6\\hfill \\\\ 4x+3y=-8\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\"><p id=\"fs-id1167834097909\">\\(\\left(-2,0,-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834448666\"><div data-type=\"problem\" id=\"fs-id1167834448668\"><p id=\"fs-id1167831824788\">\\(\\left\\{\\begin{array}{c}2x+y=3\\hfill \\\\ 6x+3y=9\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834511117\"><div data-type=\"problem\" id=\"fs-id1167835380593\"><p id=\"fs-id1167835380595\">\\(\\left\\{\\begin{array}{c}x-4y=-1\\hfill \\\\ -3x+12y=3\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834279753\"><p id=\"fs-id1167835198380\">infinitely many solutions<\/p><\/div><\/div><div data-type=\"exercise\"><div data-type=\"problem\" id=\"fs-id1167835381799\"><p id=\"fs-id1167835339290\">\\(\\left\\{\\begin{array}{c}-3x-y=4\\hfill \\\\ 6x+2y=-16\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835373726\"><div data-type=\"problem\" id=\"fs-id1167835304051\"><p id=\"fs-id1167835304053\">\\(\\left\\{\\begin{array}{c}4x+3y=2\\hfill \\\\ 20x+15y=5\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835384472\"><p id=\"fs-id1167835384474\">inconsistent<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835338037\"><div data-type=\"problem\" id=\"fs-id1167832138614\"><p id=\"fs-id1167832138616\">\\(\\left\\{\\begin{array}{c}x+y-3z=-1\\hfill \\\\ y-z=0\\hfill \\\\ \\text{\u2212}x+2y=1\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835182047\"><div data-type=\"problem\" id=\"fs-id1167835361539\"><p id=\"fs-id1167835361541\">\\(\\left\\{\\begin{array}{c}2x+3y+z=12\\hfill \\\\ x+y+z=9\\hfill \\\\ 3x+4y+2z=20\\hfill \\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835233028\"><p id=\"fs-id1167835595821\">inconsistent<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835350684\"><div data-type=\"problem\" id=\"fs-id1167835350686\"><p id=\"fs-id1167835330672\">\\(\\left\\{\\begin{array}{c}3x+4y-3z=-2\\hfill \\\\ 2x+3y-z=-12\\hfill \\\\ x+y-2z=6\\hfill \\end{array}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835595999\"><div data-type=\"problem\" id=\"fs-id1167835596001\"><p id=\"fs-id1167832056877\">\\(\\left\\{\\begin{array}{c}\\hfill x-2y+3z=1\\\\ \\hfill x+y-3z=7\\\\ \\hfill 3x-4y+5z=7\\end{array}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835419656\"><p id=\"fs-id1167835419658\">infinitely many solutions<\/p><\/div><\/div><p id=\"fs-id1167831880636\"><strong data-effect=\"bold\">Solve Applications Using Determinants<\/strong><\/p><p id=\"fs-id1167835479157\">In the following exercises, determine whether the given points are collinear.<\/p><div data-type=\"exercise\" id=\"fs-id1167835479160\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834061136\"><p id=\"fs-id1167834061138\">\\(\\left(0,1\\right),\\)\\(\\left(2,0\\right),\\) and \\(\\left(-2,2\\right).\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835594906\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835216894\"><p id=\"fs-id1167835216896\">\\(\\left(0,-5\\right),\\)\\(\\left(-2,-2\\right),\\) and \\(\\left(2,-8\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834432218\"><p id=\"fs-id1167834432220\">yes<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835369186\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834252714\"><p id=\"fs-id1167834252717\">\\(\\left(4,-3\\right),\\)\\(\\left(6,-4\\right),\\) and \\(\\left(2,-2\\right).\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826997297\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835369200\"><p id=\"fs-id1167835369202\">\\(\\left(-2,1\\right),\\)\\(\\left(-4,4\\right),\\) and \\(\\left(0,-2\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831883455\"><p id=\"fs-id1167831883457\">no<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835181788\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167826880150\"><div data-type=\"problem\" id=\"fs-id1167835361504\"><p id=\"fs-id1167835361506\">Explain the difference between a square matrix and its determinant. Give an example of each.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835381763\"><div data-type=\"problem\" id=\"fs-id1167835381765\"><p id=\"fs-id1167835341254\">Explain what is meant by the minor of an entry in a square matrix.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834191834\"><p id=\"fs-id1167834191836\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834065724\"><div data-type=\"problem\" id=\"fs-id1167834065726\"><p id=\"fs-id1167834079227\">Explain how to decide which row or column you will use to expand a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835333796\"><div data-type=\"problem\" id=\"fs-id1167835333799\"><p id=\"fs-id1167831887863\">Explain the steps for solving a system of equations using Cramer\u2019s rule.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831879999\"><p id=\"fs-id1167831880001\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834228167\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167834432226\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167832134028\" data-alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I ca, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: Evaluate the Determinant of a 2 by 2 Matrix, Evaluate the Determinant of a 3 by 3 Matrix, Use Cramer\u2019s Rule to Solve Systems of Equations, Solve Applications Using Determinants. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I ca, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: Evaluate the Determinant of a 2 by 2 Matrix, Evaluate the Determinant of a 3 by 3 Matrix, Use Cramer\u2019s Rule to Solve Systems of Equations, Solve Applications Using Determinants. The remaining columns are blank.\"><\/span><p id=\"fs-id1167835371530\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl><dt>determinant<\/dt><dd id=\"fs-id1167835254636\">Each square matrix has a real number associated with it called its determinant.<\/dd><\/dl><dl id=\"fs-id1167834517583\"><dt>minor of an entry in a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant<\/dt><dd id=\"fs-id1167834156744\">The minor of an entry in a \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant is the \\(2\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}2\\) determinant found by eliminating the row and column in the \\(3\\phantom{\\rule{0.2em}{0ex}}\u00d7\\phantom{\\rule{0.2em}{0ex}}3\\) determinant that contains the entry.<\/dd><\/dl><dl id=\"fs-id1167832055147\"><dt>square matrix<\/dt><dd id=\"fs-id1167831959509\">A square matrix is a matrix with the same number of rows and columns.<\/dd><\/dl><\/div>\n","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Evaluate the determinant of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> matrix<\/li>\n<li>Evaluate the determinant of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> matrix<\/li>\n<li>Use Cramer\u2019s Rule to solve systems of equations<\/li>\n<li>Solve applications using determinants<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" class=\"be-prepared\">\n<p id=\"fs-id1167826995899\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835531480\" type=\"1\">\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25e1090a81119426f67aae8c1e9de860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"139\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834556092\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-48ea3985af951dd31fafc87e0e01416d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#45;&#49;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#45;&#52;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"336\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536158\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Simplify: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8749e146e84ee2cc6025964788fd9af0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#45;&#49;&#50;&#125;&#123;&#45;&#56;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"30\" style=\"vertical-align: -6px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536325\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167835421208\">In this section we will learn of another method to solve systems of linear equations called Cramer\u2019s rule. Before we can begin to use the rule, we need to learn some new definitions and notation.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834517668\">\n<h3 data-type=\"title\">Evaluate the Determinant of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> Matrix<\/h3>\n<p id=\"fs-id1167835287688\">If a matrix has the same number of rows and columns, we call it a <span data-type=\"term\">square matrix<\/span>. Each square matrix has a real number associated with it called its <span data-type=\"term\">determinant<\/span>. To find the determinant of the square matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8789fe4b68113abfb5607f814dadaf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"81\" style=\"vertical-align: -17px;\" \/> we first write it as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6eda5c9dbe01085e4614d3d2eaaf5d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"72\" style=\"vertical-align: -11px;\" \/> To get the real number value of the determinate we subtract the products of the diagonals, as shown.<\/p>\n<p><span data-type=\"media\" data-alt=\"A 2 by 2 determinant is show, with its first row being a, b and second one being c, d. These values are written between two vertical lines instead of brackets as in the case of matrices. Two arrows are shown, one from a to d, the other from c to b. This determinant is equal to ad minus bc.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_001_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 2 by 2 determinant is show, with its first row being a, b and second one being c, d. These values are written between two vertical lines instead of brackets as in the case of matrices. Two arrows are shown, one from a to d, the other from c to b. This determinant is equal to ad minus bc.\" \/><\/span><\/p>\n<div data-type=\"note\" id=\"fs-id1167835374978\">\n<div data-type=\"title\">Determinant<\/div>\n<p id=\"fs-id1167834532264\">The determinant of any square matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8789fe4b68113abfb5607f814dadaf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"81\" style=\"vertical-align: -17px;\" \/> where <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em> are real numbers, is<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835301633\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-283ab048dbbea365f2f35e44964c3b56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#61;&#97;&#100;&#45;&#98;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"148\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167835193012\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832058394\">\n<div data-type=\"problem\" id=\"fs-id1167835369363\">\n<p id=\"fs-id1167834432835\">Evaluate the determinate of <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd77decd8a5ae427751809c2b440e3b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"93\" style=\"vertical-align: -17px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-febac92f3105c06d5b015957bb58d516_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832042643\">\n<p id=\"fs-id1167834191139\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167834462960\" class=\"unnumbered unstyled\" summary=\"Row 1 of the 2 by 2 matrix is 4, minus 2. Row 2 is 3, minus 1. The same is written in determinant form with diagonal arrows. Subtracting the products of the diagonals, we get 4 times minus 1 minus 3 times minus 2. We simplify to get 2.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"bottom\" data-align=\"left\">Write the determinant.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832074051\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract the products of the diagonals.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830704243\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834309193\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831846606\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_002e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835282892\" class=\"unnumbered unstyled\" summary=\"Row 1 of the 2 by 2 matrix is minus 3, minus 4. Row 2 is minus 2, 0. The same is written in determinant form with diagonal arrows. Subtracting the products of the diagonals, we get minus 3 times 0 minus minus 2 times minus 4. We simplify to get minus 8.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"bottom\" data-align=\"left\">Write the determinant.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832066568\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Subtract the products of the diagonals.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826997005\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835226257\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835253835\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835280362\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834536041\">\n<div data-type=\"problem\" id=\"fs-id1167835422303\">\n<p id=\"fs-id1167832057952\">Evaluate the determinate of <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-124ed58a949078efde1acc0c8066c4d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"93\" style=\"vertical-align: -17px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ecab14c2a2827728c27381c770f762a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835208370\">\n<p id=\"fs-id1167831912980\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1cb4591fdf1e95fa32f81f21a5eba8c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;&#59;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"35\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7eaefbab9f26b55d0afa7f1689372c40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832055377\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834373486\">\n<div data-type=\"problem\" id=\"fs-id1167827902584\">\n<p id=\"fs-id1167834357131\">Evaluate the determinate of <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84be772f477b3742ef4de0a0723dea8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"85\" style=\"vertical-align: -17px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-625ed3fd4ad046475d983aa07bc8bd95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826938221\">\n<p id=\"fs-id1167834423524\"><span class=\"token\">\u24d0<\/span> 2 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0329d19b932005c996d83dbb9e09ffed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835350146\">\n<h3 data-type=\"title\">Evaluate the Determinant of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> Matrix<\/h3>\n<p id=\"fs-id1167835498954\">To evaluate the determinant of a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> matrix, we have to be able to evaluate the <span data-type=\"term\">minor of an entry<\/span> in the determinant. The minor of an entry is the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant found by eliminating the row and column in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant that contains the entry.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834124754\">\n<div data-type=\"title\">Minor of an entry in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> a Determinant<\/div>\n<p id=\"fs-id1167832065959\">The <strong data-effect=\"bold\">minor of an entry<\/strong> in a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant is the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant found by eliminating the row and column in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant that contains the entry.<\/p>\n<\/div>\n<p id=\"fs-id1167835338688\">To find the minor of entry <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd6d5e05e9022f1ee8855acce1c293c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: -4px;\" \/> we eliminate the row and column which contain it. So we eliminate the first row and first column. Then we write the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant that remains.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834423592\" data-alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. a1 is highlighted. Lines strike out the first row and the first column. What remains is called minor of a1. It is shown as a separate determinant whose first row is b2, c2 and second row is b3, c3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_004_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. a1 is highlighted. Lines strike out the first row and the first column. What remains is called minor of a1. It is shown as a separate determinant whose first row is b2, c2 and second row is b3, c3.\" \/><\/span><\/p>\n<p id=\"fs-id1167835306351\">To find the minor of entry <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0437832b6be65671e9c217421378b6ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"19\" style=\"vertical-align: -4px;\" \/> we eliminate the row and column that contain it. So we eliminate the 2<sup>nd<\/sup> row and 2<sup>nd<\/sup> column. Then we write the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant that remains.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832124422\" data-alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. b2 is highlighted. Lines strike out the second row and second column. What remains is minor of b2. It is written as a separate determinant whose first row is a1, c1 and second row is a3, c3.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_005_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The first row of the 3 by 3 determinant is a1, b1, c1. Row 2 is a2, b2, c2. Row 3 is a3, b3, c3. b2 is highlighted. Lines strike out the second row and second column. What remains is minor of b2. It is written as a separate determinant whose first row is a1, c1 and second row is a3, c3.\" \/><\/span><\/p>\n<div data-type=\"example\" id=\"fs-id1167832052293\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167831887447\">\n<div data-type=\"problem\" id=\"fs-id1167830769653\">\n<p>For the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-43bb56ac47e4413970d527430cc705a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"193\" style=\"vertical-align: -23px;\" \/> find and then evaluate the minor of <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0c7045efae914075095c8f81de89c60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bd63c43937558e00910cf54b3f318f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834238985\">\n<p id=\"fs-id1167830954156\"><span class=\"token\">\u24d0<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835166304\" class=\"unnumbered unstyled\" summary=\"The first row of the 3 by 3 determinant is 4, minus 2, 3. Row 2 is 1, 0, minus 3. Row 3 is minus 2, minus 4, 2. Eliminating the row and column containing a1, we get the minor of a1. This 2 by 2 determinant has row 1: 0, minus 3 and row 2: minus 4, 2. Evaluate and simplify to get minus 12.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834502674\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"bottom\" data-align=\"left\">Eliminate the row and column that contains <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0526020c87962a8a5a61ef3f77d6349c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835254642\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant that remains.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831890624\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834048843\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834099451\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d1<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table class=\"unnumbered unstyled\" summary=\"The 3 by 3 matrix has row 1: 4, minus 2, 3, row 2: 1, 0, minus 3 and row 3: minus 2, minus 4, 2. Eliminating the row and column containing b3, we get minor of b3 with row 1: 4, 3 and row 2: 1, minus 3. Evaluate and simplify to get minus 15.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"bottom\" data-align=\"left\">Eliminate the row and column that contains <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f0f1c2a93db3f3d45de416526a5b01f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831893602\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant that remains.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835342689\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834474237\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"newline\"><\/div>\n<div data-type=\"newline\"><\/div>\n<p><span class=\"token\">\u24d2<\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<table id=\"fs-id1167835303369\" class=\"unnumbered unstyled can-break\" summary=\"The 3 by 3 matrix has row 1: 4, minus 2, 3, row 2: 1, 0, minus 3 and row 3: minus 2, minus 4, 2. Eliminating the row and column containing c2, we get minor of c2 with row 1: 4, minus 2 and row 2: minus 2, 4. Evaluate and simplify to get 12\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831883033\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"bottom\" data-align=\"left\">Eliminate the row and column that contains <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bd63c43937558e00910cf54b3f318f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-valign=\"bottom\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832058516\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant that remains.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835348052\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826778954\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167830865964\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834423084\">\n<div data-type=\"problem\" id=\"fs-id1167834098196\">\n<p id=\"fs-id1167835310588\">For the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b34de9c6b6d21710f4e4da9a7cc62f04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"193\" style=\"vertical-align: -22px;\" \/> find and then evaluate the minor of <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e68b6a17be4bff6b5b70a5b58144849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0ce8edc0ec3232cd1ece0932a566d7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834429082\">\n<p><span class=\"token\">\u24d0<\/span> 3 <span class=\"token\">\u24d1<\/span> 11 <span class=\"token\">\u24d2<\/span> 2<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831037153\">\n<div data-type=\"problem\" id=\"fs-id1167835367305\">\n<p id=\"fs-id1167834238946\">For the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce9c1bd30f4bb62c593ab43fbf3aa44e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"193\" style=\"vertical-align: -23px;\" \/> find and then evaluate the minor of <span class=\"token\">\u24d0<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69331bd254e20d8b9bf0dab7a436b3cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0c7045efae914075095c8f81de89c60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -3px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bd63c43937558e00910cf54b3f318f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"19\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834532254\">\n<p id=\"fs-id1167834340084\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><span class=\"token\">\u24d1<\/span> 2 <span class=\"token\">\u24d2<\/span> 3<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835352856\">We are now ready to evaluate a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant. To do this we expand by minors, which allows us to evaluate the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinants\u2014which we already know how to evaluate!<\/p>\n<p id=\"fs-id1167835326066\">To evaluate a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant by expanding by minors along the first row, we use the following pattern:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834449187\" data-alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_009_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\" \/><\/span><\/p>\n<p id=\"fs-id1167835422374\">Remember, to find the minor of an entry we eliminate the row and column that contains the entry.<\/p>\n<div data-type=\"note\" id=\"fs-id1167834397786\">\n<div data-type=\"title\">Expanding by Minors along the First Row to Evaluate a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> Determinant<\/div>\n<p id=\"fs-id1167830865570\">To evaluate a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant by <strong data-effect=\"bold\">expanding by minors along the first row<\/strong>, the following pattern:<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835234815\" data-alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_021_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\" \/><\/span><\/div>\n<div data-type=\"example\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835231574\">\n<div data-type=\"problem\" id=\"fs-id1167834179990\">\n<p id=\"fs-id1167835336341\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9dfded9a19b53f34478a0fc2fb2d1e0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: -23px;\" \/> by expanding by minors along the first row.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<table id=\"fs-id1167826808290\" class=\"unnumbered unstyled\" summary=\"The first row of the determinant is 2, minus 3, minus 1. Row 2 is 3, 2, 0. Row 3 is minus 1, minus 1, minus 2. Expanding by minors, we get 2 times minor of 2 minus 3 times minor of 3 plus minus 1 times minor of minus 1. Evaluating each determinant and simplifying, we get minus 25.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835336567\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Expand by minors along the first row<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835369420\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate each determinant.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834196236\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835305817\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830696631\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835328645\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835418958\">\n<div data-type=\"problem\" id=\"fs-id1167834523938\">\n<p id=\"fs-id1167834376483\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a72a2a00b7aeab66f01d1efd52f5bd5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"179\" style=\"vertical-align: -23px;\" \/> by expanding by minors along the first row.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831025428\">\n<p id=\"fs-id1167835300734\">37<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834387531\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835354006\">\n<div data-type=\"problem\" id=\"fs-id1167835214250\">\n<p id=\"fs-id1167835530640\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f1b60fb1fa6837bcf6afab257c46ceb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"193\" style=\"vertical-align: -23px;\" \/> by expanding by minors along the first row.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834438855\">\n<p id=\"fs-id1167835361021\">7<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>To evaluate a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.<\/p>\n<p id=\"fs-id1167830697049\">When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.<\/p>\n<div data-type=\"equation\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fc465310f4a156f5ceae03dc6190a25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"125\" style=\"vertical-align: -24px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1167834383296\">\n<div data-type=\"title\">Sign Pattern<\/div>\n<p id=\"fs-id1167830698084\">When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835310672\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fc465310f4a156f5ceae03dc6190a25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"125\" style=\"vertical-align: -24px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167834429687\">Notice that the sign pattern in the first row matches the signs between the terms in the expansion by the first row.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167834555240\" data-alt=\"A 3 by 3 determinant has row 1: plus, minus, plus, row 2: minus, plus, minus and row 3: plus, minus, plus. The three signs in the first row each point to a minor determinant in the expansion of a 3 by 3 determinant. Plus points to minor of a1, minus to the minor of b1 and plus to the minor of c1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_011_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant has row 1: plus, minus, plus, row 2: minus, plus, minus and row 3: plus, minus, plus. The three signs in the first row each point to a minor determinant in the expansion of a 3 by 3 determinant. Plus points to minor of a1, minus to the minor of b1 and plus to the minor of c1.\" \/><\/span><\/p>\n<p id=\"fs-id1167835367678\">Since we can expand by any row or column, how do we decide which row or column to use? Usually we try to pick a row or column that will make our calculation easier. If the determinant contains a 0, using the row or column that contains the 0 will make the calculations easier.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834186369\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835533969\">\n<div data-type=\"problem\" id=\"fs-id1167834257993\">\n<p id=\"fs-id1167835350318\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45b8284f459541869ce998002eea6de3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"173\" style=\"vertical-align: -23px;\" \/> by expanding by minors.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167835331737\">To expand by minors, we look for a row or column that will make our calculations easier. Since 0 is in the second row and second column, expanding by either of those is a good choice. Since the second row has fewer negatives than the second column, we will expand by the second row.<\/p>\n<table id=\"fs-id1167832056052\" class=\"unnumbered unstyled\" summary=\"A 3 by 3 determinant has row 1 4, minus 1, minus 3, row 2: 3, 0, 2 and row 3 5, minus 4, minus 3. The second row is highlighted. Expand using the second row. Be careful of the signs. The middle row is minus, plus, minus. Expanding, we get minus 3 times minor of 3 plus 0 times minor of 0 minus 2 times minor of 2. Evaluating each minor determinant and simplifying, we get 49.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834524204\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Expand using the second row.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Be careful of the signs.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835326516\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate each determinant.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835511115\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835337580\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Add.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835420254\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_012f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826784031\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834429462\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835284588\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bea50ef8a1f65fd98741c622a4f9a040_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"173\" style=\"vertical-align: -23px;\" \/> by expanding by minors.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835213837\">\n<p id=\"fs-id1167826849510\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2981688f5b26fd15fa08c897afa741ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835366417\" class=\"try\">\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835216554\">\n<p>Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46af6244165349f5e576694299824c53_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: -23px;\" \/> by expanding by minors.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831833355\">\n<p id=\"fs-id1167826799095\">8<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834431184\">\n<h3 data-type=\"title\">Use Cramer\u2019s Rule to Solve Systems of Equations<\/h3>\n<p id=\"fs-id1167835173740\">Cramer\u2019s Rule is a method of solving systems of equations using determinants. It can be derived by solving the general form of the systems of equations by elimination. Here we will demonstrate the rule for both systems of two equations with two variables and for systems of three equations with three variables.<\/p>\n<p id=\"fs-id1167835328055\">Let\u2019s start with the systems of two equations with two variables.<\/p>\n<div data-type=\"note\" id=\"fs-id1167831180556\">\n<div data-type=\"title\">Cramer\u2019s Rule for Solving a System of Two Equations<\/div>\n<p>For the system of equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c0114286fe3945369d8da597fb66c3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#120;&#43;&#123;&#98;&#125;&#95;&#123;&#49;&#125;&#121;&#61;&#123;&#107;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#120;&#43;&#123;&#98;&#125;&#95;&#123;&#50;&#125;&#121;&#61;&#123;&#107;&#125;&#95;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"146\" style=\"vertical-align: -17px;\" \/> the solution <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> can be determined by<\/p>\n<p><span data-type=\"media\" data-alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_013_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants\" \/><\/span><\/div>\n<p id=\"fs-id1167834099327\">Notice that to form the determinant <em data-effect=\"italics\">D<\/em>, we use take the coefficients of the variables.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167830838415\" data-alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is D with row 1: a1, b1 and row 2: a2, b2. Column 1 has coefficients of x and column 2 has coefficients of\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_014_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is D with row 1: a1, b1 and row 2: a2, b2. Column 1 has coefficients of x and column 2 has coefficients of\" \/><\/span><\/p>\n<p>Notice that to form the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-635a2d8e5550fce1c71492006892bbb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15bf296afb980f923c794fed350e505a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -6px;\" \/> we substitute the constants for the coefficients of the variable we are finding.<\/p>\n<p><span data-type=\"media\" data-alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is Dx has row 1: k1, b1 and row 2: k2, b2. Here columns 1 and 2 re constants and coefficients of y respectively. Determinant Dy has row 1: a1, k1 and row 2: a2, k2. Here, columns 1 and 2 are coefficients of x and constants respectively.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_015_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are a1x plus b1y equals k1 and a2x plus b2y equals k2. Here, a1, a2, b1, b2 are coefficients. The determinant is Dx has row 1: k1, b1 and row 2: k2, b2. Here columns 1 and 2 re constants and coefficients of y respectively. Determinant Dy has row 1: a1, k1 and row 2: a2, k2. Here, columns 1 and 2 are coefficients of x and constants respectively.\" \/><\/span><\/p>\n<div data-type=\"example\" id=\"fs-id1167834137622\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Solve a System of Equations Using Cramer\u2019s Rule<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835225995\">\n<div data-type=\"problem\" id=\"fs-id1167826782724\">\n<p id=\"fs-id1167834526137\">Solve using Cramer\u2019s Rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e72e961b06b6db125077c8562cdf6231_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"137\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835629178\"><span data-type=\"media\" id=\"fs-id1167826782312\" data-alt=\"The equations are 2x plus y equals minus 4 and 3x minus 2y equals minus 6. Step 1. Evaluate the determinant D, using the coefficients of the variables. Determinant D has row 1: 2, 1 and row 2: 3, minus 2. So, D is minus 7.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016a_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"The equations are 2x plus y equals minus 4 and 3x minus 2y equals minus 6. Step 1. Evaluate the determinant D, using the coefficients of the variables. Determinant D has row 1: 2, 1 and row 2: 3, minus 2. So, D is minus 7.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835215728\" data-alt=\"Step 2. Evaluate the determinant Dx. Use the constants in place of the x coefficients. We replace the coefficients of x, 2 and 3, with the constants, negative 4 and negative 6. We get Dx equal to 14.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016b_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2. Evaluate the determinant Dx. Use the constants in place of the x coefficients. We replace the coefficients of x, 2 and 3, with the constants, negative 4 and negative 6. We get Dx equal to 14.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835274917\" data-alt=\"Step 3. Evaluate the determinant Dy. Use the constants in place of the y coefficients. We replace the coefficients of y, 1 and 2, with the constants, negative 4 and negative 6. We get Dy equal to 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016c_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3. Evaluate the determinant Dy. Use the constants in place of the y coefficients. We replace the coefficients of y, 1 and 2, with the constants, negative 4 and negative 6. We get Dy equal to 0.\" \/><\/span><span data-type=\"media\" data-alt=\"Step 4. Find x and y. Substituting values of D, Dx and Dy in the equations x equal to Dx upon D and y equal to Dy upon D, we get x equal to minus 2 and y equal to 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016d_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4. Find x and y. Substituting values of D, Dx and Dy in the equations x equal to Dx upon D and y equal to Dy upon D, we get x equal to minus 2 and y equal to 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167835419109\" data-alt=\"Step 5. Write the solution as an ordered pair minus 2, 0.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016e_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5. Write the solution as an ordered pair minus 2, 0.\" \/><\/span><span data-type=\"media\" id=\"fs-id1167834396909\" data-alt=\"Step 6. Check that the ordered pair is a solution to both original equations.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_016f_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6. Check that the ordered pair is a solution to both original equations.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835389726\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831115647\">\n<div data-type=\"problem\" id=\"fs-id1167835267209\">\n<p id=\"fs-id1167835264828\">Solve using Cramer\u2019s rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-322e5244ab8bb44da372b8d7164b568b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831922888\">\n<p id=\"fs-id1167834279222\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f91e70f2b8e54719011cdc7b6ed53259_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#55;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#125;&#123;&#55;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"69\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834065903\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834161540\">\n<div data-type=\"problem\" id=\"fs-id1167835264028\">\n<p id=\"fs-id1167835511096\">Solve using Cramer\u2019s rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-45c67c25b55dfe560b02b1274f821eba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835423282\">\n<p id=\"fs-id1167834279427\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd6f8f312ab67ad0422a4959540654f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834431139\" class=\"howto\">\n<div data-type=\"title\">Solve a system of two equations using Cramer\u2019s rule.<\/div>\n<ol id=\"fs-id1167828401841\" type=\"1\" class=\"stepwise\">\n<li>Evaluate the determinant <em data-effect=\"italics\">D<\/em>, using the coefficients of the variables.<\/li>\n<li>Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21efe877e2519e327d7253c2a21ee054_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: -3px;\" \/> Use the constants in place of the <em data-effect=\"italics\">x<\/em> coefficients.<\/li>\n<li>Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f03df1831300321b8f54660b5e3926e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -6px;\" \/> Use the constants in place of the <em data-effect=\"italics\">y<\/em> coefficients.<\/li>\n<li>Find <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ae1238d59a7b701fbd1abe6b2dc88c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#125;&#123;&#68;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02e577baf95ff162e1028448f5e6a745_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#125;&#123;&#68;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"54\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the ordered pair is a solution to both original equations.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1167826778974\">To solve a system of three equations with three variables with Cramer\u2019s Rule, we basically do what we did for a system of two equations. However, we now have to solve for three variables to get the solution. The determinants are also going to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> which will make our work more interesting!<\/p>\n<div data-type=\"note\">\n<div data-type=\"title\">Cramer\u2019s Rule for Solving a System of Three Equations<\/div>\n<p id=\"fs-id1167835319418\">For the system of equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d210c81c8b094c59751f3fd4ea05947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#120;&#43;&#123;&#98;&#125;&#95;&#123;&#49;&#125;&#121;&#43;&#123;&#99;&#125;&#95;&#123;&#49;&#125;&#122;&#61;&#123;&#107;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#120;&#43;&#123;&#98;&#125;&#95;&#123;&#50;&#125;&#121;&#43;&#123;&#99;&#125;&#95;&#123;&#50;&#125;&#122;&#61;&#123;&#107;&#125;&#95;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#97;&#125;&#95;&#123;&#51;&#125;&#120;&#43;&#123;&#98;&#125;&#95;&#123;&#51;&#125;&#121;&#43;&#123;&#99;&#125;&#95;&#123;&#51;&#125;&#122;&#61;&#123;&#107;&#125;&#95;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"193\" style=\"vertical-align: -28px;\" \/> the solution <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d846d38c62afb12b0640750a85b1cb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#44;&#122;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"56\" style=\"vertical-align: -4px;\" \/> can be determined by<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167830865955\" data-alt=\"x is Dx upon D, y is Dy upon D and z is Dz upon D, where D is determinant with row 1: a1, b1, c1, row 2: a2, b2, c2, row 3: a3, b3, c3, use coefficients of the variables; Dx is determinant with row 1: k1, b1, c1, row 2: k2, b2, c2 and rwo 3: k3, b3, c3, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1, c1, row 2: a2, k2, c2 and row 3: a3, k3, c3, replace the y coefficients with constants; Dz is determinant with row 1: a1, b1, k1; row 2: a2, b2, k2, row 3: a3, b3, k3; replace the z coefficients with constants.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_017_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is Dx upon D, y is Dy upon D and z is Dz upon D, where D is determinant with row 1: a1, b1, c1, row 2: a2, b2, c2, row 3: a3, b3, c3, use coefficients of the variables; Dx is determinant with row 1: k1, b1, c1, row 2: k2, b2, c2 and rwo 3: k3, b3, c3, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1, c1, row 2: a2, k2, c2 and row 3: a3, k3, c3, replace the y coefficients with constants; Dz is determinant with row 1: a1, b1, k1; row 2: a2, b2, k2, row 3: a3, b3, k3; replace the z coefficients with constants.\" \/><\/span><\/div>\n<div data-type=\"example\" id=\"fs-id1167835240630\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834279802\">\n<div data-type=\"problem\" id=\"fs-id1167835416019\">\n<p id=\"fs-id1167835345728\">Solve the system of equations using Cramer\u2019s Rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-842316191440f6fe58d0cfe0932a7234_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#53;&#121;&#43;&#52;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#50;&#121;&#43;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"164\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835233369\">\n<table id=\"fs-id1167835311187\" class=\"unnumbered unstyled can-break\" summary=\"Evaluate the determinant D. D has row 1: 3, minus 5, 4. Row 2 is 5, 2, 1. Row 3 is 2, 3, minus 2. Expand by minors using column 1. Column 1 has the signs plus minus plus. D is 3 times first minor minus 5 times second minor plus 2 times third minor where the first minor has row 1: 2, 1 and row 2: 3, minus 2; second minor has row 1: minus 5, 4 and row 2: 3, minus 2; third minor has row 1: minus 5, 4 and row 2: 2, 1. Evaluate the determinants and simplify to get D equal to minus 37. To evaluate the determinant Dx, use the constants to replace the coefficients of x. Expand by minors using column 1. Evaluate and simplify to get Dx equal to minus 74. To evaluate the determinant Dy, use the constants to replace the coefficients of y. Expand by minors using column 2. Evaluate and simplify to get Dy equal to 111. To evaluate the determinant Dz, use the constants to replace the coefficients of z. Expand by minors using column 3. Evaluate and simplify to get Dz equal to 148. Find x, y, z and write the ordered triple 2, minus 3, minus 4. Check.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinant <em data-effect=\"italics\">D<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826801782\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Expand by minors using column 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835504018\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835235110\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834279238\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832152001\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826938247\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834131323\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21efe877e2519e327d7253c2a21ee054_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: -3px;\" \/> Use the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>constants to replace the coefficients of <em data-effect=\"italics\">x<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Expand by minors using column 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835634609\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831825107\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018l_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834556769\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018m_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834517442\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018n_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f03df1831300321b8f54660b5e3926e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -6px;\" \/> Use the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>constants to replace the coefficients of <em data-effect=\"italics\">y<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835376143\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018o_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834408455\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167826798788\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018p_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835335956\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018q_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831959318\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018r_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835327732\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018s_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835259317\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018t_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b73e68a6c36baa62aa87577338844f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#122;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"26\" style=\"vertical-align: -3px;\" \/> Use the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>constants to replace the coefficients of <em data-effect=\"italics\">z<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835423151\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018u_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834587839\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018v_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835367199\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018w_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834423610\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018x_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835325481\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018y_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835237771\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018z_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em>, and <em data-effect=\"italics\">z<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830697629\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018aa_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018bb_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832052669\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018cc_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Write the solution as an ordered triple.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835330774\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_018dd_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"bottom\" data-align=\"left\">Check that the ordered triple is a solution<\/p>\n<div data-type=\"newline\"><\/div>\n<p>to <strong data-effect=\"bold\">all three<\/strong> original equations.<\/td>\n<td data-valign=\"bottom\" data-align=\"left\">We leave the check to you.<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The solution is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7a13a2a3d0c18e5ae16486481cea9986_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835167506\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835358592\">\n<div data-type=\"problem\" id=\"fs-id1167835192469\">\n<p id=\"fs-id1167835323641\">Solve the system of equations using Cramer\u2019s Rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4a5e5a785a1993a5a27d4ea6b36501a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#56;&#121;&#43;&#50;&#122;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#45;&#51;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#50;&#121;&#45;&#50;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"178\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835363688\">\n<p id=\"fs-id1167831893121\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49aafdcaea82f239fa596920ebc02643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#44;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834433665\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835308933\">\n<div data-type=\"problem\" id=\"fs-id1167826783812\">\n<p id=\"fs-id1167835421498\">Solve the system of equations using Cramer\u2019s Rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-14f18472476237b6881448d2296bbf01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#121;&#45;&#54;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#54;&#121;&#43;&#51;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#50;&#121;&#45;&#51;&#122;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"178\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834183850\">\n<p id=\"fs-id1167835165883\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc34d64b9b18b3a1dc17dc1dcb6b5018_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835344810\">Cramer\u2019s rule does not work when the value of the <em data-effect=\"italics\">D<\/em> determinant is 0, as this would mean we would be dividing by 0. But when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-817882a4478a0c2196e958a1d46938c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/> the system is either inconsistent or dependent.<\/p>\n<p id=\"fs-id1167835310719\">When the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba1d3d148c096e3d054691f57ce75774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0a7d56d4829798fe49c8e019105296c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#44;&#123;&#68;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd642a4908d1a1815d9e1f9420a7e8c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: -3px;\" \/> are all zero, the system is consistent and dependent and there are infinitely many solutions.<\/p>\n<p id=\"fs-id1167831040599\">When the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba1d3d148c096e3d054691f57ce75774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0a7d56d4829798fe49c8e019105296c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#44;&#123;&#68;&#125;&#95;&#123;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"53\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd642a4908d1a1815d9e1f9420a7e8c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#122;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: -3px;\" \/> are not all zero, the system is inconsistent and there is no solution.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835253988\">\n<div data-type=\"title\">Dependent and Inconsistent Systems of Equations<\/div>\n<p id=\"fs-id1167835325469\">For any system of equations, where the <strong data-effect=\"bold\">value of the determinant<\/strong> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-817882a4478a0c2196e958a1d46938c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f62715934fc612d09794c15921f906e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#97;&#108;&#117;&#101;&#32;&#111;&#102;&#32;&#100;&#101;&#116;&#101;&#114;&#109;&#105;&#110;&#97;&#110;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#121;&#112;&#101;&#32;&#111;&#102;&#32;&#115;&#121;&#115;&#116;&#101;&#109;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#68;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#44;&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#122;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#114;&#101;&#32;&#97;&#108;&#108;&#32;&#122;&#101;&#114;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#110;&#115;&#105;&#115;&#116;&#101;&#110;&#116;&#32;&#97;&#110;&#100;&#32;&#100;&#101;&#112;&#101;&#110;&#100;&#101;&#110;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#102;&#105;&#110;&#105;&#116;&#101;&#108;&#121;&#32;&#109;&#97;&#110;&#121;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#68;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#44;&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#122;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#114;&#101;&#32;&#110;&#111;&#116;&#32;&#97;&#108;&#108;&#32;&#122;&#101;&#114;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#99;&#111;&#110;&#115;&#105;&#115;&#116;&#101;&#110;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"802\" style=\"vertical-align: -28px;\" \/><\/div>\n<p id=\"fs-id1168757812384\">In the next example, we will use the values of the determinants to find the solution of the system.<\/p>\n<div data-type=\"example\" id=\"fs-id1167834537639\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167827943773\">\n<div data-type=\"problem\" id=\"fs-id1167834094605\">\n<p id=\"fs-id1167834246685\">Solve the system of equations using Cramer\u2019s rule : <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69206878eb00dce3d33ef1b0a20d330c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#51;&#121;&#61;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#120;&#45;&#54;&#121;&#61;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"137\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835306172\">\n<p id=\"fs-id1167831923781\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-854255b87339b172bc6259bda0a56759_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#51;&#121;&#61;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#50;&#120;&#45;&#54;&#121;&#61;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#118;&#97;&#108;&#117;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#116;&#101;&#114;&#109;&#105;&#110;&#97;&#110;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#68;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#101;&#102;&#102;&#105;&#99;&#105;&#101;&#110;&#116;&#115;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#68;&#61;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#68;&#61;&#45;&#54;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#68;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"125\" width=\"545\" style=\"vertical-align: -55px;\" \/><\/p>\n<p id=\"fs-id1167831881762\">We cannot use Cramer\u2019s Rule to solve this system. But by looking at the value of the determinants <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-635a2d8e5550fce1c71492006892bbb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-15bf296afb980f923c794fed350e505a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -6px;\" \/> we can determine whether the system is dependent or inconsistent.<\/p>\n<p id=\"fs-id1167832058741\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fb9f7715a2e841831f220f4037719c17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#118;&#97;&#108;&#117;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#101;&#116;&#101;&#114;&#109;&#105;&#110;&#97;&#110;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#61;&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#61;&#45;&#50;&#52;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#61;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"524\" style=\"vertical-align: -36px;\" \/><\/p>\n<p id=\"fs-id1167834524879\">Since all the determinants are not zero, the system is inconsistent. There is no solution.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832075354\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831835395\">\n<div data-type=\"problem\" id=\"fs-id1167834555551\">\n<p id=\"fs-id1167835194834\">Solve the system of equations using Cramer\u2019s rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b0867447431481074c0aec7c7a04f05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#51;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#56;&#120;&#45;&#54;&#121;&#61;&#49;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826941225\">\n<p id=\"fs-id1167834537917\">no solution<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835339843\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835362814\">\n<div data-type=\"problem\" id=\"fs-id1167835363740\">\n<p id=\"fs-id1167835310829\">Solve the system of equations using Cramer\u2019s rule: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95faef55b8b2714247142267358c5341_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#61;&#45;&#51;&#121;&#43;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#54;&#121;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352006\">\n<p id=\"fs-id1167835207369\">infinite solutions<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167831092182\">\n<h3 data-type=\"title\">Solve Applications using Determinants<\/h3>\n<p id=\"fs-id1167835417818\">An interesting application of determinants allows us to test if points are collinear. Three points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac5d573fa5d16cc11d413df7cac096c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a76652cb71ecde8307c41233858b7e9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#51;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> are collinear if and only if the determinant below is zero.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835623297\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b51981f98a23556cf9f3d8e3ba9a6d7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#95;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"153\" style=\"vertical-align: -26px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1167835345043\">\n<div data-type=\"title\">Test for Collinear Points<\/div>\n<p id=\"fs-id1167835190915\">Three points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac5d573fa5d16cc11d413df7cac096c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55946fbadd9a0471df519912f22239b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a76652cb71ecde8307c41233858b7e9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#51;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> are collinear if and only if<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835575471\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b51981f98a23556cf9f3d8e3ba9a6d7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#95;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"153\" style=\"vertical-align: -26px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167832075915\">We will use this property in the next example.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835337787\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835468588\">\n<div data-type=\"problem\" id=\"fs-id1167835230111\">\n<p id=\"fs-id1167831824215\">Determine whether the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2a0c24808f5df4045e8a7264d52d247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc73f50c5f69fc53203f4cbc0d1b431b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> are collinear.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832054072\">\n<table id=\"fs-id1167834494992\" class=\"unnumbered unstyled\" summary=\"The 3 by 3 determinant has the last column with all ones. Substitute the values into the determinant. We get row 1: 5, minus 5, 1, row 2: 4, minus 3, 1 and row 3: 3, minus 1, 1. Evaluate the determinant by expanding by minors using column 3. Evaluate the determinants. Simplify to get D equal to 0. The value of the determinate is 0, so the points are collinear.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834161471\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Substitute the values into the determinant.<\/p>\n<div data-type=\"newline\"><\/div>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2a0c24808f5df4045e8a7264d52d247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc73f50c5f69fc53203f4cbc0d1b431b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d66a71b8940b998e4f29f8cccda06d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167830704055\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinant by expanding<\/p>\n<div data-type=\"newline\"><\/div>\n<p>by minors using column 3.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832052912\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Evaluate the determinants.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835363686\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834382391\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835510774\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_019f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\">The value of the determinate is 0, so the<\/p>\n<div data-type=\"newline\"><\/div>\n<p>points are collinear.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826871537\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831023875\">\n<div data-type=\"problem\" id=\"fs-id1167831066155\">\n<p id=\"fs-id1167832052184\">Determine whether the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc513d9b8e0d726d668b8792f6689fd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-38c7b21dc0185ea01b015f730d88801f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c3ac38dbb39343c28a60a287dfb114b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/> are collinear.<\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167828326434\">yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832066110\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835304607\">\n<div data-type=\"problem\" id=\"fs-id1167835346877\">\n<p id=\"fs-id1167835329140\">Determine whether the points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90dba9802eaae83d87c870bef1c2f3c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e652fad883d19697a87472434313d058_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ecc91b8d5e91ea23c08fcf2ee52342c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/> are collinear.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834064114\">\n<p id=\"fs-id1167831836301\">yes<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834196491\" class=\"media-2\">\n<p id=\"fs-id1167834244267\">Access these online resources for additional instruction and practice with solving systems of linear inequalities by graphing.<\/p>\n<ul id=\"fs-id1167830963112\" data-display=\"block\">\n<li><a href=\"https:\/\/www.openstax.org\/l\/37syslinineqgph\">Solving Systems of Linear Inequalities by Graphing<\/a><\/li>\n<li><a href=\"https:\/\/www.openstax.org\/l\/37syslineqs\">Systems of Linear Inequalities<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167831833252\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835300769\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Determinant:<\/strong> The determinant of any square matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8789fe4b68113abfb5607f814dadaf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"81\" style=\"vertical-align: -17px;\" \/> where <em data-effect=\"italics\">a, b, c,<\/em> and <em data-effect=\"italics\">d<\/em> are real numbers, is\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167830704326\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-283ab048dbbea365f2f35e44964c3b56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#98;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#99;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#61;&#97;&#100;&#45;&#98;&#99;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"148\" style=\"vertical-align: -11px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">Expanding by Minors along the First Row to Evaluate a 3 \u00d7 3 Determinant:<\/strong> To evaluate a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant by expanding by minors along the first row, the following pattern:\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167835303268\" data-alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_022_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"A 3 by 3 determinant is equal to a1 times minor of a1 minus b1 times minor of b1 plus c1 times minor of c1.\" \/><\/span> <\/li>\n<li><strong data-effect=\"bold\">Sign Pattern:<\/strong> When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167834132743\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5fc465310f4a156f5ceae03dc6190a25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"125\" style=\"vertical-align: -24px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">Cramer\u2019s Rule:<\/strong> For the system of equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c0114286fe3945369d8da597fb66c3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#120;&#43;&#123;&#98;&#125;&#95;&#123;&#49;&#125;&#121;&#61;&#123;&#107;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#120;&#43;&#123;&#98;&#125;&#95;&#123;&#50;&#125;&#121;&#61;&#123;&#107;&#125;&#95;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"146\" style=\"vertical-align: -17px;\" \/> the solution <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> can be determined by\n<div data-type=\"newline\"><\/div>\n<p> <span data-type=\"media\" id=\"fs-id1167832015822\" data-alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_023_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"x is Dx upon D and y is Dy upon D where D is determinant with row 1: a1, b1 and row 2 a2, b2, use coefficients of the variables; Dx is determinant with row 1: k1, b1 and row 2: k2, b2, replace the x coefficients with the consonants; Dy is determinant with row 1: a1, k1 and row 2: a2, k2, replace the y coefficients with constants.\" \/><\/span><\/p>\n<div data-type=\"newline\"><\/div>\n<p> Notice that to form the determinant <em data-effect=\"italics\">D<\/em>, we use take the coefficients of the variables.<\/li>\n<li><strong data-effect=\"bold\">How to solve a system of two equations using Cramer\u2019s rule.<\/strong>\n<ol id=\"fs-id1167835239369\" type=\"1\" class=\"stepwise\">\n<li>Evaluate the determinant <em data-effect=\"italics\">D<\/em>, using the coefficients of the variables.<\/li>\n<li>Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-21efe877e2519e327d7253c2a21ee054_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: -3px;\" \/> Use the constants in place of the <em data-effect=\"italics\">x<\/em> coefficients.<\/li>\n<li>Evaluate the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f03df1831300321b8f54660b5e3926e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"27\" style=\"vertical-align: -6px;\" \/> Use the constants in place of the <em data-effect=\"italics\">y<\/em> coefficients.<\/li>\n<li>Find <em data-effect=\"italics\">x<\/em> and <em data-effect=\"italics\">y<\/em>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ae1238d59a7b701fbd1abe6b2dc88c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#125;&#123;&#68;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -6px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bd995bc1560863306e81d5131391c1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#125;&#123;&#68;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/li>\n<li>Write the solution as an ordered pair.<\/li>\n<li>Check that the ordered pair is a solution to <strong data-effect=\"bold\">both<\/strong> original equations.<\/li>\n<li><strong data-effect=\"bold\">Dependent and Inconsistent Systems of Equations:<\/strong> For any system of equations, where the <strong data-effect=\"bold\">value of the determinant<\/strong> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-817882a4478a0c2196e958a1d46938c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;&#61;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"52\" style=\"vertical-align: -4px;\" \/>\n<div data-type=\"newline\"><\/div>\n<p> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f62715934fc612d09794c15921f906e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#97;&#108;&#117;&#101;&#32;&#111;&#102;&#32;&#100;&#101;&#116;&#101;&#114;&#109;&#105;&#110;&#97;&#110;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#121;&#112;&#101;&#32;&#111;&#102;&#32;&#115;&#121;&#115;&#116;&#101;&#109;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#68;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#44;&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#122;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#114;&#101;&#32;&#97;&#108;&#108;&#32;&#122;&#101;&#114;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#110;&#115;&#105;&#115;&#116;&#101;&#110;&#116;&#32;&#97;&#110;&#100;&#32;&#100;&#101;&#112;&#101;&#110;&#100;&#101;&#110;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#102;&#105;&#110;&#105;&#116;&#101;&#108;&#121;&#32;&#109;&#97;&#110;&#121;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#68;&#61;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#120;&#125;&#44;&#123;&#68;&#125;&#95;&#123;&#121;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#68;&#125;&#95;&#123;&#122;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#114;&#101;&#32;&#110;&#111;&#116;&#32;&#97;&#108;&#108;&#32;&#122;&#101;&#114;&#111;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#110;&#99;&#111;&#110;&#115;&#105;&#115;&#116;&#101;&#110;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#111;&#32;&#115;&#111;&#108;&#117;&#116;&#105;&#111;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"802\" style=\"vertical-align: -28px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Test for Collinear Points:<\/strong> Three points <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac5d573fa5d16cc11d413df7cac096c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-66ce153eccde5100a155e61eb73768a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"62\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a76652cb71ecde8307c41233858b7e9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#120;&#125;&#95;&#123;&#51;&#125;&#44;&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -4px;\" \/> are collinear if and only if\n<div data-type=\"newline\"><\/div>\n<div data-type=\"equation\" id=\"fs-id1167826997386\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b51981f98a23556cf9f3d8e3ba9a6d7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#123;&#120;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#95;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#123;&#120;&#125;&#95;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#95;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"153\" style=\"vertical-align: -26px;\" \/><\/div>\n<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835419241\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167832015675\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167834345575\"><strong data-effect=\"bold\">Evaluate the Determinant of a 2 \u00d7 2 Matrix<\/strong><\/p>\n<p id=\"fs-id1167834190283\">In the following exercises, evaluate the determinate of each square matrix.<\/p>\n<div data-type=\"exercise\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834403205\">\n<p id=\"fs-id1167834556820\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-44049615ac92c310114ac65ec7f5b113_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#54;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"93\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511346\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835333670\">\n<p id=\"fs-id1167835362153\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d6d227a42d311d894a7739430a6e53e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#52;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"85\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826778783\">\n<p id=\"fs-id1167835339455\">4<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834395430\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835378837\">\n<p id=\"fs-id1167835326831\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a07523af7d515e945f1611abbb12b4b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834395742\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834396530\">\n<p id=\"fs-id1167835281881\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70ad66aa9f4bb1fd4987446a5bba3a3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#93;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"120\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835378046\">\n<p id=\"fs-id1167835353456\">10<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835321872\"><strong data-effect=\"bold\">Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/strong><\/p>\n<p id=\"fs-id1167834505764\">In the following exercises, find and then evaluate the indicated minors.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835595008\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835609663\">\n<p id=\"fs-id1167835329320\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88819b19895d0873d4dd06baabf68cdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: -23px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find the minor <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e68b6a17be4bff6b5b70a5b58144849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b4282383c65cb486804fa511dfe7bf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835237643\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831833282\">\n<p id=\"fs-id1167831882030\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea4ff8f1fd1fa98427166cd0c2653c71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"187\" style=\"vertical-align: -22px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find the minor <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d475a40b209f4b36b08b4dd3abcd0f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"14\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4da37acd0e5d0b3aeee14efbcfc1202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167834182338\">\n<p id=\"fs-id1167835367765\"><span class=\"token\">\u24d0<\/span> 6 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4066673cb5834f8ce47cc6896c29963a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> <span class=\"token\">\u24d2<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834372737\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835283519\">\n<p id=\"fs-id1167835257388\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d0f175cd280deb255ba8165894acda3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: -23px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find the minor <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-69331bd254e20d8b9bf0dab7a436b3cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e68b6a17be4bff6b5b70a5b58144849_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e4da37acd0e5d0b3aeee14efbcfc1202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826994627\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834463841\">\n<p id=\"fs-id1167834512660\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e60a865f4f86f705cc73f6b05a4f7286_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"187\" style=\"vertical-align: -22px;\" \/><\/p>\n<div data-type=\"newline\"><\/div>\n<p>Find the minor <span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5da22eef5270793f5dc7c1f8b4383666_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a0c7045efae914075095c8f81de89c60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -3px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b4282383c65cb486804fa511dfe7bf1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#99;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/div>\n<div data-type=\"solution\" id=\"fs-id1167835509678\">\n<p id=\"fs-id1167835377182\"><span class=\"token\">\u24d0<\/span> 9 <span class=\"token\">\u24d1<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-470cb162cf92c55d139f4f69216225e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> <span class=\"token\">\u24d2<\/span> 8<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830865847\">In the following exercises, evaluate each determinant by expanding by minors along the first row.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834185930\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830961579\">\n<p id=\"fs-id1167835585208\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cf78ac99933873cb1b910d91aec692f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"173\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835309557\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835233219\">\n<p id=\"fs-id1167832153428\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0ebfd06a416bdaf0854940b412545786_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"187\" style=\"vertical-align: -22px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834061982\">\n<p id=\"fs-id1167832074771\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-532b4ff76dae3a1b4b7e48f5175c3220_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835370962\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835309883\">\n<p id=\"fs-id1167835319884\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27b3aa18e4fe943dd9dab313f3cf97b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#55;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835510649\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835186869\">\n<p id=\"fs-id1167832042307\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc1eb58d0c30ecbe65e373220848c80e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"173\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830962352\">\n<p id=\"fs-id1167831071407\">49<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831921120\">In the following exercises, evaluate each determinant by expanding by minors.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831919481\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834234224\">\n<p id=\"fs-id1167835496226\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-665097a0e371867293467239b5511b37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: -22px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834552380\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835417704\">\n<p id=\"fs-id1167835305417\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6cbc80602a372e3d92c312dcd18b6fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"173\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830699189\">\n<p id=\"fs-id1167834120481\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d15aa1d60ab00024197b84d5e3e3d75e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835200448\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835200450\">\n<p id=\"fs-id1167834377196\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e06677eb9a3bcec5fd9e26aee9a908f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"160\" style=\"vertical-align: -23px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835331651\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834433398\">\n<p id=\"fs-id1167834433400\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caa9fd4f47aa6a45cd73857221cb9131_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#51;&#92;&#92;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#52;&#92;&#92;&#32;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#48;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"173\" style=\"vertical-align: -22px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835416340\">\n<p>25<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835344963\"><strong data-effect=\"bold\">Use Cramer\u2019s Rule to Solve Systems of Equations<\/strong><\/p>\n<p id=\"fs-id1167835301822\">In the following exercises, solve each system of equations using Cramer\u2019s Rule.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835340758\">\n<div data-type=\"problem\" id=\"fs-id1167835340760\">\n<p id=\"fs-id1167835367493\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-045e9a4b173ce0a25e47612b3402a1b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#50;&#120;&#43;&#51;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826783768\">\n<div data-type=\"problem\" id=\"fs-id1167832226433\">\n<p id=\"fs-id1167832226435\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ed9df8a733f224c3cdc2d8a5b3e1b10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826972439\">\n<p id=\"fs-id1167834090753\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a4c313f9498c6ff1a35052ed315f2269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#44;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835595911\">\n<div data-type=\"problem\" id=\"fs-id1167835595913\">\n<p id=\"fs-id1167831890391\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49ff151ef7419336d89331d2bd5ec391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835321848\">\n<div data-type=\"problem\" id=\"fs-id1167835341103\">\n<p id=\"fs-id1167835166217\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a720fc661ea9c6c17a689b46d7c0c067_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834191310\">\n<p id=\"fs-id1167834191312\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd6f8f312ab67ad0422a4959540654f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834064990\">\n<div data-type=\"problem\" id=\"fs-id1167834191302\">\n<p id=\"fs-id1167834191305\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ed9df8a733f224c3cdc2d8a5b3e1b10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#50;&#121;&#61;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#51;&#121;&#61;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835364450\">\n<div data-type=\"problem\" id=\"fs-id1167832151802\">\n<p id=\"fs-id1167832151804\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49ff151ef7419336d89331d2bd5ec391_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#51;&#121;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"116\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832123911\">\n<p id=\"fs-id1167832056865\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f307aa6b65fb69739e6c2b7458409de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831985621\">\n<div data-type=\"problem\" id=\"fs-id1167831985623\">\n<p id=\"fs-id1167835306116\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27b3e6ad066bc733ddae7843c538f0f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#53;&#120;&#45;&#51;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"124\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835333914\">\n<div data-type=\"problem\" id=\"fs-id1167835344196\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ccd8d1a3f8aa7e63619e7075daf6d0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#56;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#53;&#121;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"125\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835166504\">\n<p id=\"fs-id1167835334618\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-403bc0cb12c4fb4e2bf022eff0167913_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#57;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"51\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826977931\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167834536120\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65a9aa5c41925f40b586b954f991e6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#54;&#120;&#45;&#53;&#121;&#43;&#50;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#121;&#45;&#52;&#122;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"156\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831911605\">\n<div data-type=\"problem\" id=\"fs-id1167831911607\">\n<p id=\"fs-id1167834133506\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46af7d0154576c52b63bbcb285e6c8e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#45;&#51;&#121;&#43;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#53;&#121;&#45;&#52;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#50;&#121;&#45;&#50;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"166\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826874676\">\n<p id=\"fs-id1167832066827\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8bf7af7c2b74294bafbdf216e77260d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#45;&#53;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835381001\">\n<div data-type=\"problem\" id=\"fs-id1167834372362\">\n<p id=\"fs-id1167834372365\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-76598839a094c81107ecd02b8cb09442_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#45;&#53;&#121;&#43;&#51;&#122;&#61;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#45;&#121;&#43;&#52;&#122;&#61;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#51;&#121;&#43;&#50;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"157\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826998295\">\n<div data-type=\"problem\" id=\"fs-id1167834222347\">\n<p id=\"fs-id1167834222349\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6756c4649ebcdecf0ecb323345eb407_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#49;&#49;&#120;&#43;&#57;&#121;&#43;&#50;&#122;&#61;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#53;&#121;&#43;&#51;&#122;&#61;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#121;&#43;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"175\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835333189\">\n<p id=\"fs-id1167835333191\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83047e0a7e798a1f70ee8b86a4967cc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835280485\">\n<div data-type=\"problem\" id=\"fs-id1167835280487\">\n<p id=\"fs-id1167835328954\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7fad776d83f482d29bfa359f801a709b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#50;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#121;&#43;&#51;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#45;&#53;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"124\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\">\n<p id=\"fs-id1167835347394\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd958c5a425944a914f86a59bb431a44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#53;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#121;&#45;&#122;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831910832\">\n<p id=\"fs-id1167831910834\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cccb83f742cb3d351a80d5b7c423823_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#51;&#44;&#50;&#44;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"68\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826978776\">\n<div data-type=\"problem\" id=\"fs-id1167826978778\">\n<p id=\"fs-id1167834423452\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bac1fd9dbb34b3d6893bb17cc3705dcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#121;&#43;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#120;&#43;&#51;&#121;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#55;&#120;&#43;&#122;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830700950\">\n<div data-type=\"problem\" id=\"fs-id1167830700953\">\n<p id=\"fs-id1167834537019\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cd407da293f78735b7b33e61c00624a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#45;&#122;&#61;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#53;&#121;&#43;&#50;&#122;&#61;&#45;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#52;&#120;&#43;&#51;&#121;&#61;&#45;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"126\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\">\n<p id=\"fs-id1167834097909\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59f33043d348b9479472c49b84c012e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"82\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448666\">\n<div data-type=\"problem\" id=\"fs-id1167834448668\">\n<p id=\"fs-id1167831824788\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80c4731a316ba331082ef53183022132_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#51;&#121;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"111\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834511117\">\n<div data-type=\"problem\" id=\"fs-id1167835380593\">\n<p id=\"fs-id1167835380595\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-49a15c99c7df16b2ded0cd01d8310e5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#45;&#52;&#121;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#45;&#51;&#120;&#43;&#49;&#50;&#121;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834279753\">\n<p id=\"fs-id1167835198380\">infinitely many solutions<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\">\n<div data-type=\"problem\" id=\"fs-id1167835381799\">\n<p id=\"fs-id1167835339290\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-769344e89e5c13cb165e69c4974b594c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#45;&#51;&#120;&#45;&#121;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#54;&#120;&#43;&#50;&#121;&#61;&#45;&#49;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"134\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835373726\">\n<div data-type=\"problem\" id=\"fs-id1167835304051\">\n<p id=\"fs-id1167835304053\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-213094ad68dae65639b41a71d70b4e72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#52;&#120;&#43;&#51;&#121;&#61;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#48;&#120;&#43;&#49;&#53;&#121;&#61;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"128\" style=\"vertical-align: -17px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835384472\">\n<p id=\"fs-id1167835384474\">inconsistent<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835338037\">\n<div data-type=\"problem\" id=\"fs-id1167832138614\">\n<p id=\"fs-id1167832138616\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25c76eb10dc5ed1d13d391f44dbfe3aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#120;&#43;&#121;&#45;&#51;&#122;&#61;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#121;&#45;&#122;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#120;&#43;&#50;&#121;&#61;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"147\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835182047\">\n<div data-type=\"problem\" id=\"fs-id1167835361539\">\n<p id=\"fs-id1167835361541\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aea6601b106cf1465bade3cf7be311ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&#120;&#43;&#51;&#121;&#43;&#122;&#61;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#43;&#122;&#61;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#43;&#50;&#122;&#61;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"161\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835233028\">\n<p id=\"fs-id1167835595821\">inconsistent<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835350684\">\n<div data-type=\"problem\" id=\"fs-id1167835350686\">\n<p id=\"fs-id1167835330672\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f45a19c3eea57ce3b02e4bc58e777033_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#51;&#120;&#43;&#52;&#121;&#45;&#51;&#122;&#61;&#45;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#50;&#120;&#43;&#51;&#121;&#45;&#122;&#61;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#120;&#43;&#121;&#45;&#50;&#122;&#61;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"165\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835595999\">\n<div data-type=\"problem\" id=\"fs-id1167835596001\">\n<p id=\"fs-id1167832056877\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94ce9f9713cc48dd79a71a0d98c7be4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#45;&#50;&#121;&#43;&#51;&#122;&#61;&#49;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#43;&#121;&#45;&#51;&#122;&#61;&#55;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#120;&#45;&#52;&#121;&#43;&#53;&#122;&#61;&#55;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"152\" style=\"vertical-align: -28px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835419656\">\n<p id=\"fs-id1167835419658\">infinitely many solutions<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831880636\"><strong data-effect=\"bold\">Solve Applications Using Determinants<\/strong><\/p>\n<p id=\"fs-id1167835479157\">In the following exercises, determine whether the given points are collinear.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835479160\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834061136\">\n<p id=\"fs-id1167834061138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-672d8db2238417510b5142a21411e47a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9bfe5f0e85bf2ee8d0ce478041861f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18cf428adb33cae2d0a4e2f88efa4d62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835594906\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835216894\">\n<p id=\"fs-id1167835216896\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b38ffa7bcb6973cafa01d99b72c88f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8712c4d601b412ff52a62ba6886fca65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"73\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1853a8ff040a07b6fa970b59332a455c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834432218\">\n<p id=\"fs-id1167834432220\">yes<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835369186\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834252714\">\n<p id=\"fs-id1167834252717\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cc73f50c5f69fc53203f4cbc0d1b431b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#44;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1fff9ddd5f00145c8bf07fb731c8c489_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-534204d07513941720df89b471c28252_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826997297\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835369200\">\n<p id=\"fs-id1167835369202\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eab971bf2865462054d1a3708e7a3bc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0af282153e9cfce96860aa3e3b9052cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f644d689ddf634b480d4ff320ff89c7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831883455\">\n<p id=\"fs-id1167831883457\">no<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167835181788\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167826880150\">\n<div data-type=\"problem\" id=\"fs-id1167835361504\">\n<p id=\"fs-id1167835361506\">Explain the difference between a square matrix and its determinant. Give an example of each.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835381763\">\n<div data-type=\"problem\" id=\"fs-id1167835381765\">\n<p id=\"fs-id1167835341254\">Explain what is meant by the minor of an entry in a square matrix.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834191834\">\n<p id=\"fs-id1167834191836\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834065724\">\n<div data-type=\"problem\" id=\"fs-id1167834065726\">\n<p id=\"fs-id1167834079227\">Explain how to decide which row or column you will use to expand a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835333796\">\n<div data-type=\"problem\" id=\"fs-id1167835333799\">\n<p id=\"fs-id1167831887863\">Explain the steps for solving a system of equations using Cramer\u2019s rule.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831879999\">\n<p id=\"fs-id1167831880001\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167834228167\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167834432226\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167832134028\" data-alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I ca, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: Evaluate the Determinant of a 2 by 2 Matrix, Evaluate the Determinant of a 3 by 3 Matrix, Use Cramer\u2019s Rule to Solve Systems of Equations, Solve Applications Using Determinants. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2018\/12\/CNX_IntAlg_Figure_04_06_201_img_new.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has 4 columns, 4 rows and a header row. The header row labels each column: I ca, confidently, with some help and no, I don\u2019t get it. The first column has the following statements: Evaluate the Determinant of a 2 by 2 Matrix, Evaluate the Determinant of a 3 by 3 Matrix, Use Cramer\u2019s Rule to Solve Systems of Equations, Solve Applications Using Determinants. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167835371530\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl>\n<dt>determinant<\/dt>\n<dd id=\"fs-id1167835254636\">Each square matrix has a real number associated with it called its determinant.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167834517583\">\n<dt>minor of an entry in a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant<\/dt>\n<dd id=\"fs-id1167834156744\">The minor of an entry in a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant is the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e168e1cb19a9b7191510ddd9eb34ece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"24\" style=\"vertical-align: 0px;\" \/> determinant found by eliminating the row and column in the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b4f8995101fede235cd5ea21a1ec3ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&times;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\" \/> determinant that contains the entry.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167832055147\">\n<dt>square matrix<\/dt>\n<dd id=\"fs-id1167831959509\">A square matrix is a matrix with the same number of rows and columns.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-2666","chapter","type-chapter","status-publish","hentry"],"part":2305,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2666","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2666\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/2305"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/2666\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=2666"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=2666"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=2666"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=2666"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}