{"id":4100,"date":"2022-04-14T18:15:54","date_gmt":"2022-04-14T18:15:54","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-change-of-variables-in-multiple-integrals\/"},"modified":"2022-11-09T16:42:01","modified_gmt":"2022-11-09T16:42:01","slug":"skills-review-for-change-of-variables-in-multiple-integrals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/skills-review-for-change-of-variables-in-multiple-integrals\/","title":{"raw":"Skills Review for Change of Variables in Multiple Integrals","rendered":"Skills Review for Change of Variables in Multiple Integrals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Evaluate the determinant of a 2\u00d72 Matrix<\/li>\r\n \t<li>Evaluate the determinant of a 3\u00d73 Matrix<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn the Change of Variables in Multiple Integrals section, we will explore how to use change of variables to calculate double and triple integrals. Here we will review how to evaluate the determinant of 2 x 2 and 3 x 3 matrices.\r\n<h2>Evaluate the Determinant of a 2\u00d72 Matrix<\/h2>\r\n<strong><em>(also in Module 2, Skills Review for the Dot Product, Cross Product, and Equations of Lines and Planes in Space)<\/em><\/strong>\r\nA determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.\r\n<div class=\"textbox shaded\">\r\n<h3>A General Note: Find the Determinant of a 2 \u00d7 2 Matrix<\/h3>\r\nThe <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex]<\/div>\r\nis defined as\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/>\r\n\r\nNotice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\r\nFind the determinant of the given matrix.\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"149250\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"149250\"]\r\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&amp;=\\left\\rvert\\begin{array}{cc}5&amp; 2\\\\ -6&amp; 3\\end{array}\\right\\rvert\\hfill \\\\ &amp;=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &amp;=27\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]6397[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/h2>\r\nFinding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.\r\n\r\nFind the <strong>determinant<\/strong> of the 3\u00d73 matrix.\r\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right][\/latex]<\/div>\r\n<ol>\r\n \t<li>Augment [latex]A[\/latex] with the first two columns.\r\n<div style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}{a}_{1}&amp; {b}_{1}&amp; {c}_{1}\\\\ {a}_{2}&amp; {b}_{2}&amp; {c}_{2}\\\\ {a}_{3}&amp; {b}_{3}&amp; {c}_{3}\\end{array}\\right\\rvert \\left.\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}\\right\\rvert[\/latex]<\/div><\/li>\r\n \t<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\r\n \t<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\r\n<\/ol>\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181433\/CNX_Precalc_Figure_09_08_0022.jpg\" alt=\"\" width=\"487\" height=\"89\" \/>\r\n\r\nThe algebra is as follows:\r\n<div style=\"text-align: center;\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\r\n<div><\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Determinant of a 3 \u00d7 3 Matrix<\/h3>\r\nFind the determinant of the given 3 \u00d7 3 matrix:\r\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right][\/latex]<\/p>\r\n[reveal-answer q=\"915069\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"915069\"]\r\n\r\nAugment the matrix with the first two columns and then follow the formula. Thus,\r\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&amp;=\\left\\rvert\\begin{array}{ccc}0&amp; 2&amp; 1\\\\ 3&amp; -1&amp; 1\\\\ 4&amp; 0&amp; 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 &amp; 2\\\\ 3 &amp; -1\\\\ 4 &amp; 0\\end{array}\\right\\rvert\\hfill \\\\ &amp;=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &amp;=0+8+0+4 - 0-6\\hfill \\\\ &amp;=6\\hfill \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\nFind the determinant of the 3 \u00d7 3 matrix.\r\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}1&amp; -3&amp; 7\\\\ 1&amp; 1&amp; 1\\\\ 1&amp; -2&amp; 3\\end{array}\\right\\rvert[\/latex]<\/p>\r\n[reveal-answer q=\"612489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"612489\"]\r\n\r\n[latex]-10[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question hide_question_numbers=1]19398[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Evaluate the determinant of a 2\u00d72 Matrix<\/li>\n<li>Evaluate the determinant of a 3\u00d73 Matrix<\/li>\n<\/ul>\n<\/div>\n<p>In the Change of Variables in Multiple Integrals section, we will explore how to use change of variables to calculate double and triple integrals. Here we will review how to evaluate the determinant of 2 x 2 and 3 x 3 matrices.<\/p>\n<h2>Evaluate the Determinant of a 2\u00d72 Matrix<\/h2>\n<p><strong><em>(also in Module 2, Skills Review for the Dot Product, Cross Product, and Equations of Lines and Planes in Space)<\/em><\/strong><br \/>\nA determinant is a real number that can be very useful in mathematics because it has multiple applications, such as calculating area, volume, and other quantities. Here, we will use determinants to reveal whether a matrix is invertible by using the entries of a <strong>square matrix<\/strong> to determine whether there is a solution to the system of equations. Perhaps one of the more interesting applications, however, is their use in cryptography. Secure signals or messages are sometimes sent encoded in a matrix. The data can only be decrypted with an <strong>invertible matrix<\/strong> and the determinant. For our purposes, we focus on the determinant as an indication of the invertibility of the matrix. Calculating the determinant of a matrix involves following the specific patterns that are outlined in this section.<\/p>\n<div class=\"textbox shaded\">\n<h3>A General Note: Find the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p>The <strong>determinant<\/strong> of a [latex]2\\text{ }\\times \\text{ }2[\/latex] matrix, given<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex]<\/div>\n<p>is defined as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181431\/CNX_Precalc_Figure_09_08_0012.jpg\" alt=\"\" width=\"487\" height=\"59\" \/><\/p>\n<p>Notice the change in notation. There are several ways to indicate the determinant, including [latex]\\mathrm{det}\\left(A\\right)[\/latex] and replacing the brackets in a matrix with straight lines, [latex]|A|[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Determinant of a 2 \u00d7 2 Matrix<\/h3>\n<p>Find the determinant of the given matrix.<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q149250\">Show Solution<\/span><\/p>\n<div id=\"q149250\" class=\"hidden-answer\" style=\"display: none\">\n<p style=\"text-align: center;\">[latex]\\begin{align}\\mathrm{det}\\left(A\\right)&=\\left\\rvert\\begin{array}{cc}5& 2\\\\ -6& 3\\end{array}\\right\\rvert\\hfill \\\\ &=5\\left(3\\right)-\\left(-6\\right)\\left(2\\right)\\hfill \\\\ &=27\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm6397\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=6397&theme=oea&iframe_resize_id=ohm6397\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Evaluate the Determinant of a 3 \u00d7 3 Matrix<\/h2>\n<p>Finding the determinant of a 2\u00d72 matrix is straightforward, but finding the determinant of a 3\u00d73 matrix is more complicated. One method is to augment the 3\u00d73 matrix with a repetition of the first two columns, giving a 3\u00d75 matrix. Then we calculate the sum of the products of entries <em>down<\/em> each of the three diagonals (upper left to lower right), and subtract the products of entries <em>up<\/em> each of the three diagonals (lower left to upper right). This is more easily understood with a visual and an example.<\/p>\n<p>Find the <strong>determinant<\/strong> of the 3\u00d73 matrix.<\/p>\n<div style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right][\/latex]<\/div>\n<ol>\n<li>Augment [latex]A[\/latex] with the first two columns.\n<div style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}{a}_{1}& {b}_{1}& {c}_{1}\\\\ {a}_{2}& {b}_{2}& {c}_{2}\\\\ {a}_{3}& {b}_{3}& {c}_{3}\\end{array}\\right\\rvert \\left.\\begin{array}{c}{a}_{1}\\\\ {a}_{2}\\\\ {a}_{3}\\end{array}\\begin{array}{c}{b}_{1}\\\\ {b}_{2}\\\\ {b}_{3}\\end{array}\\right\\rvert[\/latex]<\/div>\n<\/li>\n<li>From upper left to lower right: Multiply the entries down the first diagonal. Add the result to the product of entries down the second diagonal. Add this result to the product of the entries down the third diagonal.<\/li>\n<li>From lower left to upper right: Subtract the product of entries up the first diagonal. From this result subtract the product of entries up the second diagonal. From this result, subtract the product of entries up the third diagonal.<\/li>\n<\/ol>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27181433\/CNX_Precalc_Figure_09_08_0022.jpg\" alt=\"\" width=\"487\" height=\"89\" \/><\/p>\n<p>The algebra is as follows:<\/p>\n<div style=\"text-align: center;\">[latex]|A|={a}_{1}{b}_{2}{c}_{3}+{b}_{1}{c}_{2}{a}_{3}+{c}_{1}{a}_{2}{b}_{3}-{a}_{3}{b}_{2}{c}_{1}-{b}_{3}{c}_{2}{a}_{1}-{c}_{3}{a}_{2}{b}_{1}[\/latex]<\/div>\n<div><\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Determinant of a 3 \u00d7 3 Matrix<\/h3>\n<p>Find the determinant of the given 3 \u00d7 3 matrix:<\/p>\n<p style=\"text-align: center;\">[latex]A=\\left[\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right][\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q915069\">Show Solution<\/span><\/p>\n<div id=\"q915069\" class=\"hidden-answer\" style=\"display: none\">\n<p>Augment the matrix with the first two columns and then follow the formula. Thus,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{align}|A|&=\\left\\rvert\\begin{array}{ccc}0& 2& 1\\\\ 3& -1& 1\\\\ 4& 0& 1\\end{array}\\right\\rvert\\left.\\begin{array}{c}0 & 2\\\\ 3 & -1\\\\ 4 & 0\\end{array}\\right\\rvert\\hfill \\\\ &=0\\left(-1\\right)\\left(1\\right)+2\\left(1\\right)\\left(4\\right)+1\\left(3\\right)\\left(0\\right)-4\\left(-1\\right)\\left(1\\right)-0\\left(1\\right)\\left(0\\right)-1\\left(3\\right)\\left(2\\right)\\hfill \\\\ &=0+8+0+4 - 0-6\\hfill \\\\ &=6\\hfill \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p>Find the determinant of the 3 \u00d7 3 matrix.<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{det}\\left(A\\right)=\\left\\rvert\\begin{array}{ccc}1& -3& 7\\\\ 1& 1& 1\\\\ 1& -2& 3\\end{array}\\right\\rvert[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q612489\">Show Solution<\/span><\/p>\n<div id=\"q612489\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]-10[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm19398\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=19398&theme=oea&iframe_resize_id=ohm19398\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-4100\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 1. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus1\/\">https:\/\/courses.lumenlearning.com\/calculus1\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Calculus Volume 2. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/calculus2\/\">https:\/\/courses.lumenlearning.com\/calculus2\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 1\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/calculus1\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/calculus2\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4100","chapter","type-chapter","status-publish","hentry"],"part":4135,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4100","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4100\/revisions"}],"predecessor-version":[{"id":6489,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4100\/revisions\/6489"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/4135"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/4100\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=4100"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=4100"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=4100"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=4100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}