{"id":234464,"date":"2025-06-14T06:44:46","date_gmt":"2025-06-14T06:44:46","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=234464"},"modified":"2025-06-14T06:44:48","modified_gmt":"2025-06-14T06:44:48","slug":"how-many-free-variables-does-each-augmented-matrix-have","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/14\/how-many-free-variables-does-each-augmented-matrix-have\/","title":{"rendered":"How many free variables does each augmented matrix have"},"content":{"rendered":"\n<p> How many free variables does each augmented matrix have? One % s] Three None Two<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/image-478.png\" alt=\"\" class=\"wp-image-234465\"\/><\/figure>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-0-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Here are the correct answers and a detailed explanation for each augmented matrix.<\/p>\n\n\n\n<p><strong>Correct Answers:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>a. Two<\/strong><\/li>\n\n\n\n<li><strong>b. Two<\/strong><\/li>\n\n\n\n<li><strong>c. One<\/strong><\/li>\n\n\n\n<li><strong>d. None<\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>In linear algebra, an augmented matrix represents a system of linear equations. The variables in this system can be classified as either&nbsp;<strong>basic variables<\/strong>&nbsp;or&nbsp;<strong>free variables<\/strong>. To determine the number of free variables, we first need to identify the pivot positions in the matrix, which is assumed to be in row echelon form.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A\u00a0<strong>pivot position<\/strong>\u00a0in a matrix is the location of a leading &#8216;1&#8217; in a non-zero row after the matrix has been brought to reduced row echelon form. In a matrix already in echelon form, it is the first non-zero entry in each non-zero row.<\/li>\n\n\n\n<li>A\u00a0<strong>pivot column<\/strong>\u00a0is a column that contains a pivot position.<\/li>\n\n\n\n<li>A\u00a0<strong>basic variable<\/strong>\u00a0is a variable that corresponds to a pivot column.<\/li>\n\n\n\n<li>A\u00a0<strong>free variable<\/strong>\u00a0is any variable that does not correspond to a pivot column. The number of free variables is therefore the total number of variables (columns in the coefficient matrix) minus the number of basic variables (pivot columns). Free variables can be assigned any value, and the values of the basic variables will depend on them.<\/li>\n<\/ul>\n\n\n\n<p>Let&#8217;s analyze each matrix:<\/p>\n\n\n\n<p><strong>a.<\/strong>&nbsp;The augmented matrix is:&nbsp;[ 1 -2 -10 | -5 ]<br>[ 0 0 0 | 0 ]<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Identify Variables:<\/strong>\u00a0The coefficient matrix (the part before the vertical line) has 3 columns. Let&#8217;s call the variables x\u2081, x\u2082, and x\u2083.<\/li>\n\n\n\n<li><strong>Identify Pivot Columns:<\/strong>\u00a0The first non-zero entry in the first row is in the\u00a0<strong>first column<\/strong>. The second row is all zeros, so it has no pivot. Therefore, only the first column is a pivot column.<\/li>\n\n\n\n<li><strong>Count Free Variables:<\/strong>\u00a0Since column 1 is a pivot column, x\u2081 is a basic variable. Columns 2 and 3 do not contain a pivot, so x\u2082 and x\u2083 are free variables.\n<ul class=\"wp-block-list\">\n<li><strong>Number of free variables: Two<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p><strong>b.<\/strong>&nbsp;The augmented matrix is:&nbsp;[ 1 0 0 -10 | 6 ]<br>[ 0 1 0 0 | 10 ]<br>[ 0 0 0 0 | 0 ]<br>[ 0 0 0 0 | 0 ]<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Identify Variables:<\/strong>\u00a0The coefficient matrix has 4 columns (x\u2081, x\u2082, x\u2083, x\u2084).<\/li>\n\n\n\n<li><strong>Identify Pivot Columns:<\/strong>\u00a0The pivot in the first row is in the\u00a0<strong>first column<\/strong>. The pivot in the second row is in the\u00a0<strong>second column<\/strong>. The other rows are zero rows.<\/li>\n\n\n\n<li><strong>Count Free Variables:<\/strong>\u00a0Columns 1 and 2 are pivot columns, so x\u2081 and x\u2082 are basic variables. Columns 3 and 4 do not contain pivots, so x\u2083 and x\u2084 are free variables.\n<ul class=\"wp-block-list\">\n<li><strong>Number of free variables: Two<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p><strong>c.<\/strong>&nbsp;The augmented matrix is:&nbsp;[ 1 -2 | 2 ]<br>[ 0 0 | 0 ]<br>[ 0 0 | 0 ]<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Identify Variables:<\/strong>\u00a0The coefficient matrix has 2 columns (x\u2081, x\u2082).<\/li>\n\n\n\n<li><strong>Identify Pivot Columns:<\/strong>\u00a0The pivot in the first row is in the\u00a0<strong>first column<\/strong>. The other rows are zero rows.<\/li>\n\n\n\n<li><strong>Count Free Variables:<\/strong>\u00a0Column 1 is a pivot column, so x\u2081 is a basic variable. Column 2 does not contain a pivot, so x\u2082 is a free variable.\n<ul class=\"wp-block-list\">\n<li><strong>Number of free variables: One<\/strong><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p><strong>d.<\/strong>&nbsp;The augmented matrix is:&nbsp;[ 1 0 | -8 ]<br>[ 0 1 | -4 ]<br>[ 0 0 | 0 ]<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Identify Variables:<\/strong>\u00a0The coefficient matrix has 2 columns (x\u2081, x\u2082).<\/li>\n\n\n\n<li><strong>Identify Pivot Columns:<\/strong>\u00a0The pivot in the first row is in the\u00a0<strong>first column<\/strong>. The pivot in the second row is in the\u00a0<strong>second column<\/strong>.<\/li>\n\n\n\n<li><strong>Count Free Variables:<\/strong>\u00a0Both column 1 and column 2 are pivot columns. Thus, both x\u2081 and x\u2082 are basic variables. There are no columns left without a pivot. This system has a unique solution (x\u2081 = -8, x\u2082 = -4).\n<ul class=\"wp-block-list\">\n<li><strong>Number of free variables: None<\/strong>\u00a0(or zero)<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-415.jpeg\" alt=\"\" class=\"wp-image-234466\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>How many free variables does each augmented matrix have? One % s] Three None Two The Correct Answer and Explanation is: Here are the correct answers and a detailed explanation for each augmented matrix. Correct Answers: Explanation In linear algebra, an augmented matrix represents a system of linear equations. The variables in this system can [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-234464","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234464","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=234464"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/234464\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=234464"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=234464"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=234464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}