{"id":1822,"date":"2015-11-12T18:30:44","date_gmt":"2015-11-12T18:30:44","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1822"},"modified":"2015-11-12T18:30:44","modified_gmt":"2015-11-12T18:30:44","slug":"key-concepts-glossary-27","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/key-concepts-glossary-27\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<h2>Key Concepts<\/h2>\n<ul><li>The determinant for [latex]\\left[\\begin{array}{cc}a&amp; b\\\\ c&amp; d\\end{array}\\right][\/latex] is [latex]ad-bc[\/latex].<\/li>\n\t<li>Cramer\u2019s Rule replaces a variable column with the constant column. Solutions are [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D}[\/latex].<\/li>\n\t<li>To find the determinant of a 3\u00d73 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right).<\/li>\n\t<li>To solve a system of three equations in three variables using Cramer\u2019s Rule, replace a variable column with the constant column for each desired solution: [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D}[\/latex].<\/li>\n\t<li>Cramer\u2019s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions.<\/li>\n\t<li>Certain properties of determinants are useful for solving problems. For example:\n<ul><li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n\t<li>When two rows are interchanged, the determinant changes sign.<\/li>\n\t<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n\t<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n\t<li>The determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant of the matrix [latex]A[\/latex].<\/li>\n\t<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\n<\/ul><\/li>\n<\/ul><h2>Glossary<\/h2>\n<dl id=\"fs-id1674058\" class=\"definition\"><dt>Cramer\u2019s Rule<\/dt><dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd><\/dl><dl id=\"fs-id1674068\" class=\"definition\"><dt>determinant<\/dt><dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd><\/dl>\u00a0","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>The determinant for [latex]\\left[\\begin{array}{cc}a& b\\\\ c& d\\end{array}\\right][\/latex] is [latex]ad-bc[\/latex].<\/li>\n<li>Cramer\u2019s Rule replaces a variable column with the constant column. Solutions are [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D}[\/latex].<\/li>\n<li>To find the determinant of a 3\u00d73 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right).<\/li>\n<li>To solve a system of three equations in three variables using Cramer\u2019s Rule, replace a variable column with the constant column for each desired solution: [latex]x=\\frac{{D}_{x}}{D},y=\\frac{{D}_{y}}{D},z=\\frac{{D}_{z}}{D}[\/latex].<\/li>\n<li>Cramer\u2019s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions.<\/li>\n<li>Certain properties of determinants are useful for solving problems. For example:\n<ul>\n<li>If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.<\/li>\n<li>When two rows are interchanged, the determinant changes sign.<\/li>\n<li>If either two rows or two columns are identical, the determinant equals zero.<\/li>\n<li>If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.<\/li>\n<li>The determinant of an inverse matrix [latex]{A}^{-1}[\/latex] is the reciprocal of the determinant of the matrix [latex]A[\/latex].<\/li>\n<li>If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1674058\" class=\"definition\">\n<dt>Cramer\u2019s Rule<\/dt>\n<dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd>\n<\/dl>\n<dl id=\"fs-id1674068\" class=\"definition\">\n<dt>determinant<\/dt>\n<dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd>\n<\/dl>\n<p>\u00a0<\/p>\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-1822\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/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":276,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"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-1822","chapter","type-chapter","status-publish","hentry"],"part":1811,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1822","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/users\/276"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1822\/revisions"}],"predecessor-version":[{"id":2235,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1822\/revisions\/2235"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/1811"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/1822\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/media?parent=1822"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=1822"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/contributor?post=1822"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/wp-json\/wp\/v2\/license?post=1822"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}