{"id":2275,"date":"2016-11-03T19:47:52","date_gmt":"2016-11-03T19:47:52","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2275"},"modified":"2017-08-15T21:25:25","modified_gmt":"2017-08-15T21:25:25","slug":"introduction-gaussian-elimination","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/introduction-gaussian-elimination\/","title":{"raw":"Introduction to Gaussian Elimination","rendered":"Introduction to Gaussian Elimination"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Write the augmented matrix of a system of equations.<\/li>\r\n \t<li>Write the system of equations from an augmented matrix.<\/li>\r\n \t<li>Perform row operations on a matrix.<\/li>\r\n \t<li>Solve a system of linear equations using matrices.<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_3207\" align=\"aligncenter\" width=\"235\"]<img class=\"size-medium wp-image-3207\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/20224012\/CNX_Precalc_Figure_09_06_001new2-235x300.jpg\" alt=\"painting of Gauss\" width=\"235\" height=\"300\" \/> German mathematician Carl Friedrich Gauss (1777\u20131855).[\/caption]\r\n\r\nCarl Friedrich <strong>Gauss<\/strong> lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.\r\n\r\nWe first encountered Gaussian elimination in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/chapter\/introduction-systems-of-linear-equations-two-variables\/\" target=\"_blank\" rel=\"noopener\">Systems of Linear Equations: Two Variables<\/a>. In this section, we will revisit this technique for solving systems, this time using matrices.","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Write the augmented matrix of a system of equations.<\/li>\n<li>Write the system of equations from an augmented matrix.<\/li>\n<li>Perform row operations on a matrix.<\/li>\n<li>Solve a system of linear equations using matrices.<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_3207\" style=\"width: 245px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-3207\" class=\"size-medium wp-image-3207\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/01\/20224012\/CNX_Precalc_Figure_09_06_001new2-235x300.jpg\" alt=\"painting of Gauss\" width=\"235\" height=\"300\" \/><\/p>\n<p id=\"caption-attachment-3207\" class=\"wp-caption-text\">German mathematician Carl Friedrich Gauss (1777\u20131855).<\/p>\n<\/div>\n<p>Carl Friedrich <strong>Gauss<\/strong> lived during the late 18th century and early 19th century, but he is still considered one of the most prolific mathematicians in history. His contributions to the science of mathematics and physics span fields such as algebra, number theory, analysis, differential geometry, astronomy, and optics, among others. His discoveries regarding matrix theory changed the way mathematicians have worked for the last two centuries.<\/p>\n<p>We first encountered Gaussian elimination in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/chapter\/introduction-systems-of-linear-equations-two-variables\/\" target=\"_blank\" rel=\"noopener\">Systems of Linear Equations: Two Variables<\/a>. In this section, we will revisit this technique for solving systems, this time using matrices.<\/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-2275\">\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":21,"menu_order":6,"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 al.\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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