{"id":156510,"date":"2022-03-26T01:50:13","date_gmt":"2022-03-25T20:20:13","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/elementary-transformation-of-matrix-introduction-definition-methods-calculation-solved-examples\/"},"modified":"2025-06-23T17:03:11","modified_gmt":"2025-06-23T11:33:11","slug":"elementary-transformation-of-matrix-introduction-definition-methods-calculation-solved-examples","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/","title":{"rendered":"Elementary Transformation of Matrix &#8211; Introduction, Definition"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#What_is_Elementary_Transformation_of_the_Matrix\" title=\"What is Elementary Transformation of the Matrix?\">What is Elementary Transformation of the Matrix?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#Elementary_Row_Transformations\" title=\"Elementary Row Transformations\">Elementary Row Transformations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#Example_for_Row_Equivalent_Matrices\" title=\"Example for Row Equivalent Matrices\">Example for Row Equivalent Matrices<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#Elementary_Column_Transformations\" title=\"Elementary Column Transformations\">Elementary Column Transformations<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#Elementary_Row_Transformations-2\" title=\"Elementary Row Transformations\">Elementary Row Transformations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#Example_for_Row_Equivalent_Matrices-2\" title=\"Example for Row Equivalent Matrices\">Example for Row Equivalent Matrices<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#Elementary_Column_Transformations-2\" title=\"Elementary Column Transformations\">Elementary Column Transformations<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/elementary-transformation-of-matrices\/#Fun_Facts\" title=\"Fun Facts\">Fun Facts<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Elementary_Transformation_of_the_Matrix\"><\/span>What is Elementary Transformation of the Matrix?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Elementary transformation of the matrix is the transformation of a matrix in which the matrix is multiplied by a scalar. The scalar can be either a real number or an imaginary number. Elementary Transformation of Matrix &#8211; Introduction Definition.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156509 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/elementary-transformation-of-matrix-introduction-definition-methods-calculation-solved-examples.jpg\" alt=\"Elementary Transformation of Matrix - Introduction, Definition\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/elementary-transformation-of-matrix-introduction-definition-methods-calculation-solved-examples.jpg?v=1648239611 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/elementary-transformation-of-matrix-introduction-definition-methods-calculation-solved-examples-300x212.jpg?v=1648239611 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Elementary_Row_Transformations\"><\/span>Elementary Row Transformations<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>There are three elementary row transformations that can be performed on a matrix A:<\/p>\n<p>1. Rotation: A rotates its rows by a certain angle \u03b8.<br \/>\n2. Reflection: A reflects its rows across a certain line y = x.<br \/>\n3. Transposition: A transposes its rows by swapping the first row with the last row.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Example_for_Row_Equivalent_Matrices\"><\/span>Example for Row Equivalent Matrices<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The row equivalent matrices are:<\/p>\n<p>A =<\/p>\n<p>B =<\/p>\n<p>C =<\/p>\n<p>D =<\/p>\n<p>The matrices are row equivalent because they have the same number of rows and the same number of columns. The corresponding elements in each matrix are also equal.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Elementary_Column_Transformations\"><\/span>Elementary Column Transformations<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>There are three basic types of column transformations:<\/p>\n<p>1. Rotations<br \/>\n2. Reflections<br \/>\n3. Shear<\/p>\n<h3 dir=\"ltr\"><span class=\"ez-toc-section\" id=\"Elementary_Row_Transformations-2\"><\/span>Elementary Row Transformations<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p dir=\"ltr\">Row transformations are performed only on the basis of a few sets of rules. An individual cannot perform any other kind of row operation apart from the below-stated rules. There are three kinds of elementary row transformations.<\/p>\n<ol>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Interchanging the rows within the matrix: In this operation, the entire row in a matrix is swapped with another row. It is symbolically represented as Ri \u2194 Rj, where i and j are two different row numbers.<\/p>\n<\/li>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Scaling the entire row with a non zero number: The entire row is multiplied with the same non zero number. It is symbolically represented as Ri \u2192 k Ri which indicates that each element of the row is scaled by a factor \u2018k\u2019.<\/p>\n<\/li>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Add one row to another row multiplied by a non zero number: Each element of a row is replaced by a number obtained by adding it to the scaled element of another row. It is symbolically represented as Ri \u2192 Ri + k Rj.<\/p>\n<\/li>\n<\/ol>\n<p dir=\"ltr\">Two matrices are said to be row equivalent if and only if one matrix can be obtained from the other by performing any of the above elementary row transformations.<\/p>\n<h3 dir=\"ltr\"><span class=\"ez-toc-section\" id=\"Example_for_Row_Equivalent_Matrices-2\"><\/span>Example for Row Equivalent Matrices<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p dir=\"ltr\">1. Show that matrices A and B are row equivalent if<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-1-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;\/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mo&gt;&amp;#x2212;&lt;\/mo&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;\/mtable&gt;&lt;mo&gt;]&lt;\/mo&gt;&lt;\/mrow&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mtext&gt;and B&lt;\/mtext&gt;&lt;\/mrow&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;\/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;\/mtable&gt;&lt;mo&gt;]&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-1\" class=\"math\"><span id=\"MathJax-Span-2\" class=\"mrow\"><span id=\"MathJax-Span-3\" class=\"mi\">A<\/span><span id=\"MathJax-Span-4\" class=\"mo\">=<\/span><span id=\"MathJax-Span-5\" class=\"mrow\"><span id=\"MathJax-Span-6\" class=\"mo\">[<\/span><span id=\"MathJax-Span-7\" class=\"mtable\"><span id=\"MathJax-Span-8\" class=\"mtd\"><span id=\"MathJax-Span-9\" class=\"mrow\"><span id=\"MathJax-Span-10\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-18\" class=\"mtd\"><span id=\"MathJax-Span-19\" class=\"mrow\"><span id=\"MathJax-Span-20\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-11\" class=\"mtd\"><span id=\"MathJax-Span-12\" class=\"mrow\"><span id=\"MathJax-Span-13\" class=\"mo\">\u2212<\/span><span id=\"MathJax-Span-14\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-21\" class=\"mtd\"><span id=\"MathJax-Span-22\" class=\"mrow\"><span id=\"MathJax-Span-23\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-15\" class=\"mtd\"><span id=\"MathJax-Span-16\" class=\"mrow\"><span id=\"MathJax-Span-17\" class=\"mn\">0<\/span><\/span><\/span><span id=\"MathJax-Span-24\" class=\"mtd\"><span id=\"MathJax-Span-25\" class=\"mrow\"><span id=\"MathJax-Span-26\" class=\"mn\">1<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-27\" class=\"mo\">]<\/span><\/span><span id=\"MathJax-Span-28\" class=\"texatom\"><span id=\"MathJax-Span-29\" class=\"mrow\"><span id=\"MathJax-Span-30\" class=\"mtext\">and B<\/span><\/span><\/span><span id=\"MathJax-Span-31\" class=\"mo\">=<\/span><span id=\"MathJax-Span-32\" class=\"mrow\"><span id=\"MathJax-Span-33\" class=\"mo\">[<\/span><span id=\"MathJax-Span-34\" class=\"mtable\"><span id=\"MathJax-Span-35\" class=\"mtd\"><span id=\"MathJax-Span-36\" class=\"mrow\"><span id=\"MathJax-Span-37\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-44\" class=\"mtd\"><span id=\"MathJax-Span-45\" class=\"mrow\"><span id=\"MathJax-Span-46\" class=\"mn\">0<\/span><\/span><\/span><span id=\"MathJax-Span-38\" class=\"mtd\"><span id=\"MathJax-Span-39\" class=\"mrow\"><span id=\"MathJax-Span-40\" class=\"mn\">0<\/span><\/span><\/span><span id=\"MathJax-Span-47\" class=\"mtd\"><span id=\"MathJax-Span-48\" class=\"mrow\"><span id=\"MathJax-Span-49\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-41\" class=\"mtd\"><span id=\"MathJax-Span-42\" class=\"mrow\"><span id=\"MathJax-Span-43\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-50\" class=\"mtd\"><span id=\"MathJax-Span-51\" class=\"mrow\"><span id=\"MathJax-Span-52\" class=\"mn\">1<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-53\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><mi>A<\/mi><mo>=<\/mo><mrow><mo>[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>1<\/mn><\/mtd><mtd><mo>\u2212<\/mo><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mn>2<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo>]<\/mo><\/mrow><mrow class=\"MJX-TeXAtom-ORD\"><mtext>and B<\/mtext><\/mrow><mo>=<\/mo><mrow><mo>[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>3<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>3<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo>]<\/mo><\/mrow><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">Solution:<\/p>\n<p dir=\"ltr\">Consider the matrix A. Apply row transformation such that R1 \u2192 R1 + R2<\/p>\n<p dir=\"ltr\">Applying row transformations to the first row, A11 = 1 + 2, A12 = -1 + 1 and A13 = 0 + 1<\/p>\n<p dir=\"ltr\">So matrix A will be equal to<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-2-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;\/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;\/mtable&gt;&lt;mo&gt;]&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-54\" class=\"math\"><span id=\"MathJax-Span-55\" class=\"mrow\"><span id=\"MathJax-Span-56\" class=\"mrow\"><span id=\"MathJax-Span-57\" class=\"mo\">[<\/span><span id=\"MathJax-Span-58\" class=\"mtable\"><span id=\"MathJax-Span-59\" class=\"mtd\"><span id=\"MathJax-Span-60\" class=\"mrow\"><span id=\"MathJax-Span-61\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-68\" class=\"mtd\"><span id=\"MathJax-Span-69\" class=\"mrow\"><span id=\"MathJax-Span-70\" class=\"mn\">2<\/span><\/span><\/span><span id=\"MathJax-Span-62\" class=\"mtd\"><span id=\"MathJax-Span-63\" class=\"mrow\"><span id=\"MathJax-Span-64\" class=\"mn\">0<\/span><\/span><\/span><span id=\"MathJax-Span-71\" class=\"mtd\"><span id=\"MathJax-Span-72\" class=\"mrow\"><span id=\"MathJax-Span-73\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-65\" class=\"mtd\"><span id=\"MathJax-Span-66\" class=\"mrow\"><span id=\"MathJax-Span-67\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-74\" class=\"mtd\"><span id=\"MathJax-Span-75\" class=\"mrow\"><span id=\"MathJax-Span-76\" class=\"mn\">1<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-77\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><mo>[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>3<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>2<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo>]<\/mo><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">Now let us retain the first row and apply row transformation to the second row such that<\/p>\n<p dir=\"ltr\">R2 \u2192 3 R2 &#8211; R1<\/p>\n<p dir=\"ltr\">So the elements of second row in A will be given as follows:<\/p>\n<p dir=\"ltr\">A21 = 2 x 3 &#8211; 3 = 3<\/p>\n<p dir=\"ltr\">A22 = 1 x 3 &#8211; 0 = 3<\/p>\n<p dir=\"ltr\">A23 = 1 x 3 &#8211; 1 = 2<\/p>\n<p dir=\"ltr\">So matrix A will be equal to<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-3-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;\/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;\/mtable&gt;&lt;mo&gt;]&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-78\" class=\"math\"><span id=\"MathJax-Span-79\" class=\"mrow\"><span id=\"MathJax-Span-80\" class=\"mrow\"><span id=\"MathJax-Span-81\" class=\"mo\">[<\/span><span id=\"MathJax-Span-82\" class=\"mtable\"><span id=\"MathJax-Span-83\" class=\"mtd\"><span id=\"MathJax-Span-84\" class=\"mrow\"><span id=\"MathJax-Span-85\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-92\" class=\"mtd\"><span id=\"MathJax-Span-93\" class=\"mrow\"><span id=\"MathJax-Span-94\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-86\" class=\"mtd\"><span id=\"MathJax-Span-87\" class=\"mrow\"><span id=\"MathJax-Span-88\" class=\"mn\">0<\/span><\/span><\/span><span id=\"MathJax-Span-95\" class=\"mtd\"><span id=\"MathJax-Span-96\" class=\"mrow\"><span id=\"MathJax-Span-97\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-89\" class=\"mtd\"><span id=\"MathJax-Span-90\" class=\"mrow\"><span id=\"MathJax-Span-91\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-98\" class=\"mtd\"><span id=\"MathJax-Span-99\" class=\"mrow\"><span id=\"MathJax-Span-100\" class=\"mn\">2<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-101\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><mo>[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>3<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>3<\/mn><\/mtd><mtd><mn>3<\/mn><\/mtd><mtd><mn>2<\/mn><\/mtd><\/mtr><\/mtable><mo>]<\/mo><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">Retain R1 and apply row transformation to R2 such  that R2 \u2192 R2 &#8211; R1.<\/p>\n<p dir=\"ltr\">A21 = 3 &#8211; 3 = 0<\/p>\n<p dir=\"ltr\">A22 = 3 &#8211; 0 = 3<\/p>\n<p dir=\"ltr\">A23 = 2 &#8211; 1 = 1<\/p>\n<p dir=\"ltr\">So the matrix A will be equal to matrix B.<\/p>\n<div class=\"MathJax_Display\">\n<p><span id=\"MathJax-Element-4-Frame\" class=\"MathJax\" style=\"box-sizing: border-box; font-family: 'Open Sans', sans-serif; -webkit-tap-highlight-color: rgba(255, 255, 255, 0); display: inline; font-style: normal; font-weight: normal; line-height: 49px; font-size: 17.3333px; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;\/mo&gt;&lt;mtable rowspacing=&quot;4pt&quot; columnspacing=&quot;1em&quot;&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;\/mtd&gt;&lt;mtd&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;\/mtd&gt;&lt;\/mtr&gt;&lt;\/mtable&gt;&lt;mo&gt;]&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-102\" class=\"math\"><span id=\"MathJax-Span-103\" class=\"mrow\"><span id=\"MathJax-Span-104\" class=\"mrow\"><span id=\"MathJax-Span-105\" class=\"mo\">[<\/span><span id=\"MathJax-Span-106\" class=\"mtable\"><span id=\"MathJax-Span-107\" class=\"mtd\"><span id=\"MathJax-Span-108\" class=\"mrow\"><span id=\"MathJax-Span-109\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-116\" class=\"mtd\"><span id=\"MathJax-Span-117\" class=\"mrow\"><span id=\"MathJax-Span-118\" class=\"mn\">0<\/span><\/span><\/span><span id=\"MathJax-Span-110\" class=\"mtd\"><span id=\"MathJax-Span-111\" class=\"mrow\"><span id=\"MathJax-Span-112\" class=\"mn\">0<\/span><\/span><\/span><span id=\"MathJax-Span-119\" class=\"mtd\"><span id=\"MathJax-Span-120\" class=\"mrow\"><span id=\"MathJax-Span-121\" class=\"mn\">3<\/span><\/span><\/span><span id=\"MathJax-Span-113\" class=\"mtd\"><span id=\"MathJax-Span-114\" class=\"mrow\"><span id=\"MathJax-Span-115\" class=\"mn\">1<\/span><\/span><\/span><span id=\"MathJax-Span-122\" class=\"mtd\"><span id=\"MathJax-Span-123\" class=\"mrow\"><span id=\"MathJax-Span-124\" class=\"mn\">1<\/span><\/span><\/span><\/span><span id=\"MathJax-Span-125\" class=\"mo\">]<\/span><\/span><\/span><\/span><\/span><\/p>\n<p><mo>[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><mn>3<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mn>3<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><\/mtable><mo>]<\/mo><\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<p dir=\"ltr\">From this, we can conclude that A and B are row equivalent matrices. Elementary Transformation of Matrix &#8211; Introduction .<\/p>\n<h3 dir=\"ltr\"><span class=\"ez-toc-section\" id=\"Elementary_Column_Transformations-2\"><\/span>Elementary Column Transformations<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p dir=\"ltr\">There are also a few sets of rules to be followed while performing column transformations. There are three different forms of elementary column transformations. No other column transformations are allowed apart from these three.<\/p>\n<ol>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Interchanging the columns within the matrix: In this operation, the entire column in a matrix is swapped with another column. It is symbolically represented as  Ci \u2194 Cj, where i and j are two different column numbers.<\/p>\n<\/li>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Multiplying the entire column with a non zero number: The entire column is multiplied or divided by the same non zero number. It is symbolically represented as Ci \u2192 k Ci which indicates that each element of the column is multiplied by a scaling factor \u2018k\u2019.<\/p>\n<\/li>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Add one column to another column scaled by a non zero number: Each element of a column is replaced by a number obtained by adding it to the scaled element of another column. It is symbolically represented as Ci \u2192 Ci + k Cj.<\/p>\n<\/li>\n<\/ol>\n<p dir=\"ltr\">Two matrices are said to be column equivalent if and only if one matrix can be obtained from the other by performing any of the above elementary column transformations. Elementary Transformation of Matrix &#8211; Introduction .<\/p>\n<h3 dir=\"ltr\"><span class=\"ez-toc-section\" id=\"Fun_Facts\"><\/span>Fun Facts<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Equal matrices have the same order and the same elements.<\/p>\n<\/li>\n<li dir=\"ltr\" aria-level=\"1\">\n<p dir=\"ltr\" role=\"presentation\">Equivalent matrices are the matrices with the same order and similar elements. Two matrices are said to be equivalent if one matrix can be obtained from the other using the idea of \u2018What is Elementary transformation\u2019.<\/p>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>What is Elementary Transformation of the Matrix? 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