{"id":335,"date":"2023-06-21T13:22:58","date_gmt":"2023-06-21T13:22:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-gaussian-elimination\/"},"modified":"2023-06-21T13:22:58","modified_gmt":"2023-06-21T13:22:58","slug":"summary-gaussian-elimination","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-gaussian-elimination\/","title":{"raw":"Summary: Gaussian Elimination","rendered":"Summary: Gaussian Elimination"},"content":{"raw":"\n\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>An augmented matrix is one that contains the coefficients and constants of a system of equations.<\/li>\n \t<li>A matrix augmented with the constant column can be represented as the original system of equations.<\/li>\n \t<li>Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.<\/li>\n \t<li>We can use Gaussian elimination to solve a system of equations.<\/li>\n \t<li>Row operations are performed on matrices to obtain row-echelon form.<\/li>\n \t<li>To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions.<\/li>\n \t<li>A calculator can be used to solve systems of equations using matrices.<\/li>\n \t<li>Many real-world problems can be solved using augmented matrices.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>augmented matrix<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt><strong>coefficient matrix<\/strong><\/dt>\n \t<dd id=\"fs-id1165133085665\">a matrix that contains only the coefficients from a system of equations<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>Gaussian elimination<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">using elementary row operations to obtain a matrix in row-echelon form<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>main diagonal<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">entries from the upper left corner diagonally to the lower right corner of a square matrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt><strong>row-echelon form<\/strong><\/dt>\n \t<dd id=\"fs-id1165133085665\">after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n \t<dt>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n \t<dt>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>row-equivalent<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">two matrices [latex]A[\/latex] and [latex]B[\/latex] are row-equivalent if one can be obtained from the other by performing basic row operations<\/dd>\n<\/dl>\n<\/dt>\n<\/dl>\n<\/dt>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>row operations<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form<\/dd>\n<\/dl>\n\n","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>An augmented matrix is one that contains the coefficients and constants of a system of equations.<\/li>\n<li>A matrix augmented with the constant column can be represented as the original system of equations.<\/li>\n<li>Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.<\/li>\n<li>We can use Gaussian elimination to solve a system of equations.<\/li>\n<li>Row operations are performed on matrices to obtain row-echelon form.<\/li>\n<li>To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions.<\/li>\n<li>A calculator can be used to solve systems of equations using matrices.<\/li>\n<li>Many real-world problems can be solved using augmented matrices.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt><strong>augmented matrix<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>coefficient matrix<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">a matrix that contains only the coefficients from a system of equations<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>Gaussian elimination<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">using elementary row operations to obtain a matrix in row-echelon form<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt>\n<\/dt>\n<dt><strong>main diagonal<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">entries from the upper left corner diagonally to the lower right corner of a square matrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt><strong>row-echelon form<\/strong><\/dt>\n<dd id=\"fs-id1165133085665\">after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133085661\" class=\"definition\">\n<dt>\n<\/dt>\n<dt><strong>row-equivalent<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">two matrices [latex]A[\/latex] and [latex]B[\/latex] are row-equivalent if one can be obtained from the other by performing basic row operations<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>row operations<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form<\/dd>\n<\/dl>\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-335\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><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":395986,"menu_order":17,"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\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\"}]","CANDELA_OUTCOMES_GUID":"53c4c098-c17a-46a1-96fe-453432857881","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-335","chapter","type-chapter","status-publish","hentry"],"part":323,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/335","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/users\/395986"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/335\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/parts\/323"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/335\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/media?parent=335"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=335"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/contributor?post=335"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/wp-json\/wp\/v2\/license?post=335"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}