{"id":13461,"date":"2023-05-01T10:00:58","date_gmt":"2023-05-01T04:30:58","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=13461"},"modified":"2023-05-02T10:13:14","modified_gmt":"2023-05-02T04:43:14","slug":"solving-systems-linear-equations-using-matrices","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/solving-systems-linear-equations-using-matrices\/","title":{"rendered":"Solving Systems of Linear Equations Using Matrices"},"content":{"rendered":"<h2>Solving Systems of Linear Equations Using Matrices<\/h2>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/NNmiOoWt86M?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Homogeneous and non-homogeneous systems of linear equations<\/h3>\n<p>A system of equations AX = B is called a homogeneous system if B = O. If B \u2260 O, it is called a non-homogeneous system of equations.<br \/>\ne.g., 2x + 5y = 0<br \/>\n3x \u2013 2y = 0<br \/>\nis a homogeneous system of linear equations whereas the system of equations given by<br \/>\ne.g., 2x + 3y = 5<br \/>\nx +\u00a0y =\u00a02<br \/>\nis a non-homogeneous system of linear equations.<\/p>\n<h3>Solution of Non-homogeneous system of linear equations<\/h3>\n<ol>\n<li>Matrix method: If AX = B, then X = A<sup>-1<\/sup>B\u00a0gives a unique solution, provided A is non-singular.<br \/>\nBut if A is a singular matrix i.e., if |A| = 0, then the system of equation AX = B may be consistent with infinitely many solutions or it may be inconsistent.<\/li>\n<li>Rank method for solution of Non-Homogeneous system AX = B\n<ol>\n<li>Write down A, B<\/li>\n<li>Write the augmented matrix [A : B]<\/li>\n<li>Reduce the augmented matrix to Echelon form by using elementary row operations.<\/li>\n<li>Find the number of non-zero rows in A and [A : B] to find the ranks of A and [A : B] respectively.<\/li>\n<li>If \u03c1(A) \u2260 \u03c1(A : B)\u00a0then the system is inconsistent.<\/li>\n<li>\u03c1(A) =\u00a0\u03c1(A : B) = the number of unknowns, then the system has a unique solution.<\/li>\n<li>\u03c1(A) =\u00a0\u03c1(A : B) &lt; number of unknowns, then the system has an infinite number of solutions.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/a2z7sZ4MSqo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<h3>Solutions of a homogeneous system of linear equations<\/h3>\n<p>Let AX = O\u00a0be a homogeneous system of 3 linear equations in 3 unknowns.<\/p>\n<ol>\n<li>Write the given system of equations in the form AX = O and write A.<\/li>\n<li>Find |A|.<\/li>\n<li>If |A| \u2260 0, then the system is consistent and x = y = z = 0 is the unique solution.<\/li>\n<li>If |A| = 0, then the systems of equations has infinitely many solutions. In order to find that put z = k (any real number) and solve any two equations for x and y so obtained with z = k give a solution of the given system of equations.<\/li>\n<\/ol>\n<h3>Consistency of a system of linear equation AX = B, where A is a square matrix<\/h3>\n<p>In system of linear equations AX = B, A = (a<sub>ij<\/sub>)<sub>n<\/sub><sub>\u00d7n<\/sub>\u00a0is said to be<\/p>\n<ol>\n<li>Consistent (with unique solution) if |A| \u2260 0.<br \/>\ni.e., if A is non-singular matrix.<\/li>\n<li>Inconsistent (It has no solution) if |A| =\u00a00\u00a0and (adj A)B is a non-null matrix.<\/li>\n<li>Consistent (with infinitely m any solutions) if |A| =\u00a00\u00a0and (adj A)B is a null matrix.<\/li>\n<\/ol>\n<h3>Rank of matrix<\/h3>\n<p><strong>Definition:<\/strong><br \/>\nLet A be a m\u00d7n matrix. If we retain any r rows and r columns of A we shall have a square sub-matrix of order r. The determinant of the square sub-matrix of order r is called a minor of A order r. Consider any matrix A which is of the order of 3\u00d74 say,<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128965\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-1.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 1\" width=\"151\" height=\"98\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-1.jpg 151w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-1-150x98.jpg 150w\" sizes=\"(max-width: 151px) 100vw, 151px\" \/>.<br \/>\nIt is 3\u00d74 matrix so we can have minors of order 3, 2 or 1. Taking any three rows and three columns minor of order three. Hence minor of order \\(3=\\left| \\begin{matrix} 1 &amp; 3 &amp; 4 \\\\ 1 &amp; 2 &amp; 6 \\\\ 1 &amp; 5 &amp; 0 \\end{matrix} \\right| =0\\)<br \/>\nMaking two zeros and expanding above minor is zero. Similarly we can consider any other minor of order 3 and it can be shown to be zero. Minor of order 2 is obtained by taking any two rows and any two columns.<br \/>\nMinor of order \\(2=\\begin{vmatrix} 1 &amp; 3 \\\\ 1 &amp; 2 \\end{vmatrix}=2-3=-1\\neq 0\\).<br \/>\nMinor of order 1 is every element of the matrix.<\/p>\n<p><strong>Rank of a matrix:<\/strong> The rank of a given matrix A is said to be r if<\/p>\n<ol>\n<li>Every minor of A of order r+1 is zero.<\/li>\n<li>There is at least one minor of A of order r which does not vanish. Here we can also say that the rank of a matrix A is said to be r ,if\n<ul>\n<li>Every square submatrix of order r+1 is singular.<\/li>\n<li>There is at least one square submatrix of order r which is non-singular.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p>The rank r of matrix A is written as \u03c1(A) = r.<\/p>\n<h3>Echelon form of a matrix<\/h3>\n<p>A matrix A is said to be in Echelon form if either A is the null matrix or A satisfies the following conditions:<\/p>\n<ol>\n<li>Every non- zero row in A precedes every zero row.<\/li>\n<li>The number of zeros before the first non-zero element in a row is less than the number of such zeros in the next row.<\/li>\n<\/ol>\n<p>If can be easily proved that the rank of a matrix in Echelon form is equal to the number of non-zero row of the matrix.<\/p>\n<p><strong>Rank of a matrix in Echelon form:<\/strong> The rank of a matrix in Echelon form is equal to the number of non-zero rows in that matrix.<\/p>\n<h3>Solving Systems of Linear Equations Using Matrices Problems with Solutions<\/h3>\n<p><strong>1.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128966\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-2.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 2\" width=\"510\" height=\"164\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-2.jpg 510w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-2-300x96.jpg 300w\" sizes=\"(max-width: 510px) 100vw, 510px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128967\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-3.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 3\" width=\"540\" height=\"155\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-3.jpg 540w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-3-300x86.jpg 300w\" sizes=\"(max-width: 540px) 100vw, 540px\" \/><br \/>\n<strong>2.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128969\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-4.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 4\" width=\"539\" height=\"195\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-4.jpg 539w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-4-300x109.jpg 300w\" sizes=\"(max-width: 539px) 100vw, 539px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128970\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-5.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 5\" width=\"545\" height=\"564\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-5.jpg 545w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-5-290x300.jpg 290w\" sizes=\"(max-width: 545px) 100vw, 545px\" \/><br \/>\n<strong>3.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128971\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-6.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 6\" width=\"483\" height=\"112\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-6.jpg 483w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-6-300x70.jpg 300w\" sizes=\"(max-width: 483px) 100vw, 483px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128972\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-7.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 7\" width=\"521\" height=\"160\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-7.jpg 521w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-7-300x92.jpg 300w\" sizes=\"(max-width: 521px) 100vw, 521px\" \/><br \/>\n<strong>4.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128973\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-8.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 8\" width=\"514\" height=\"149\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-8.jpg 514w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-8-300x87.jpg 300w\" sizes=\"(max-width: 514px) 100vw, 514px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128975\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-9.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 9\" width=\"553\" height=\"355\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-9.jpg 553w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-9-300x193.jpg 300w\" sizes=\"(max-width: 553px) 100vw, 553px\" \/><br \/>\n<strong>5.<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128976\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-10.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 10\" width=\"468\" height=\"296\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-10.jpg 468w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-10-300x190.jpg 300w\" sizes=\"(max-width: 468px) 100vw, 468px\" \/><br \/>\n<strong>Solution:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-128977\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-11.jpg\" alt=\"Solving Systems of Linear Equations Using Matrices 11\" width=\"440\" height=\"287\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-11.jpg 440w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/05\/Solving-Systems-of-Linear-Equations-Using-Matrices-11-300x196.jpg 300w\" sizes=\"(max-width: 440px) 100vw, 440px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solving Systems of Linear Equations Using Matrices Homogeneous and non-homogeneous systems of linear equations A system of equations AX = B is called a homogeneous system if B = O. If B \u2260 O, it is called a non-homogeneous system of equations. e.g., 2x + 5y = 0 3x \u2013 2y = 0 is a [&hellip;]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[4955,4957,4952,4956,4953,4954,4951],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Solving Systems of Linear Equations Using Matrices - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/solving-systems-linear-equations-using-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Solving Systems of Linear Equations Using Matrices\" \/>\n<meta property=\"og:description\" content=\"Solving Systems of Linear Equations Using Matrices Homogeneous and non-homogeneous systems of linear equations A system of equations AX = B is called a homogeneous system if B = O. 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