{"id":1801,"date":"2015-11-12T18:30:45","date_gmt":"2015-11-12T18:30:45","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&#038;p=1801"},"modified":"2015-11-12T18:30:45","modified_gmt":"2015-11-12T18:30:45","slug":"key-concepts-glossary-29","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/key-concepts-glossary-29\/","title":{"raw":"Key Concepts &amp; Glossary","rendered":"Key Concepts &amp; Glossary"},"content":{"raw":"<h2>Key Concepts<\/h2>\n<ul><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><h2>Glossary<\/h2>\n<dl id=\"fs-id1165134279527\" class=\"definition\"><dt>augmented matrix<\/dt><dd id=\"fs-id1165134279532\">a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets<\/dd><\/dl><dl id=\"fs-id1165133103268\" class=\"definition\"><dt>coefficient matrix<\/dt><dd id=\"fs-id1165133103274\">a matrix that contains only the coefficients from a system of equations<\/dd><\/dl><dl id=\"fs-id1165134245049\" class=\"definition\"><dt>Gaussian elimination<\/dt><dd id=\"fs-id1165134245055\">using elementary row operations to obtain a matrix in row-echelon form<\/dd><\/dl><dl id=\"fs-id1165134245059\" class=\"definition\"><dt>main diagonal<\/dt><dd id=\"fs-id1165134493415\">entries from the upper left corner diagonally to the lower right corner of a square matrix<\/dd><\/dl><dl id=\"fs-id1165134493421\" class=\"definition\"><dt>row-echelon form<\/dt><dd id=\"fs-id1165134486699\">after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal<\/dd><\/dl><dl id=\"fs-id1165134486705\" class=\"definition\"><dt>row-equivalent<\/dt><dd id=\"fs-id1165131866861\">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><\/dl><dl id=\"fs-id1165137456553\" class=\"definition\"><dt>row operations<\/dt><dd id=\"fs-id1165132961357\">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><\/dl>","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-id1165134279527\" class=\"definition\">\n<dt>augmented matrix<\/dt>\n<dd id=\"fs-id1165134279532\">a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133103268\" class=\"definition\">\n<dt>coefficient matrix<\/dt>\n<dd id=\"fs-id1165133103274\">a matrix that contains only the coefficients from a system of equations<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134245049\" class=\"definition\">\n<dt>Gaussian elimination<\/dt>\n<dd id=\"fs-id1165134245055\">using elementary row operations to obtain a matrix in row-echelon form<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134245059\" class=\"definition\">\n<dt>main diagonal<\/dt>\n<dd id=\"fs-id1165134493415\">entries from the upper left corner diagonally to the lower right corner of a square matrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134493421\" class=\"definition\">\n<dt>row-echelon form<\/dt>\n<dd id=\"fs-id1165134486699\">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-id1165134486705\" class=\"definition\">\n<dt>row-equivalent<\/dt>\n<dd id=\"fs-id1165131866861\">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-id1165137456553\" class=\"definition\">\n<dt>row operations<\/dt>\n<dd id=\"fs-id1165132961357\">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-1801\">\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 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