{"id":667703,"date":"2023-08-28T14:47:29","date_gmt":"2023-08-28T09:17:29","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=667703"},"modified":"2023-08-28T15:37:01","modified_gmt":"2023-08-28T10:07:01","slug":"identity-matrix","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/","title":{"rendered":"Identity Matrix"},"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\/articles\/identity-matrix\/#What_is_Identity_Matrix\" title=\"What is Identity Matrix?\">What is Identity Matrix?<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Definition_of_an_Identity_Matrix\" title=\"Definition of an Identity Matrix\">Definition of an Identity Matrix<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Characteristics_of_Identity_Matrix\" title=\"Characteristics of Identity Matrix\">Characteristics of Identity Matrix<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Types_of_Identity_Matrices\" title=\"Types of Identity Matrices\">Types of Identity Matrices<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Steps_to_check_the_given_Matrix_is_Identity_Matrix_or_not\" title=\"Steps to check the given Matrix is Identity Matrix or not\">Steps to check the given Matrix is Identity Matrix or not<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Frequently_asked_questions_on_FAQs_Identity_Matrix\" title=\"Frequently asked questions on  (FAQs) Identity Matrix\">Frequently asked questions on  (FAQs) Identity Matrix<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Can_Identity_matrix_be_zero_matrix\" title=\"Can Identity matrix be zero matrix? \">Can Identity matrix be zero matrix? <\/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\/articles\/identity-matrix\/#Is_the_Identity_matrix_equal_to_1\" title=\"Is the Identity matrix equal to 1?\">Is the Identity matrix equal to 1?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#What_is_the_shape_of_an_Identity_matrix\" title=\"What is the shape of an Identity matrix? \">What is the shape of an Identity matrix? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#What_is_the_identity_and_null_matrix\" title=\"What is the identity and null matrix ? \">What is the identity and null matrix ? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Is_there_any_3_x_2_Identity_matrix\" title=\"Is there any 3 x 2 Identity matrix? \">Is there any 3 x 2 Identity matrix? <\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/articles\/identity-matrix\/#Are_Identity_matrices_being_only_square_matrices\" title=\"Are Identity matrices being only square matrices \">Are Identity matrices being only square matrices <\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"What_is_Identity_Matrix\"><\/span>What is Identity Matrix?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>An identity matrix (I or In) is a square matrix with ones on its major diagonal (from top left to bottom right) and zeros everywhere else. on other words, except for the entries on the major diagonal, all the elements in the identity matrix are zero. An identity matrix has the same number of rows as columns and is sometimes denoted by the subscript &#8220;n&#8221; (for example, I3 for a 3&#215;3 identity matrix). When a square matrix is multiplied by its matching identity matrix, the original matrix stays intact, much like when an integer is multiplied by 1. Identity matrices are useful in linear algebra, matrix operations, and the solution of systems of equations.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Definition_of_an_Identity_Matrix\"><\/span>Definition of an Identity Matrix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>An identity matrix, abbreviated as I or In, is a square matrix with all major diagonal members being ones (1) and all other components being zeros (0). It has the feature of returning the original matrix when multiplied with any suitable matrix. In other words, the identity matrix acts in <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/matrix-multiplication\/\"><strong>matrix multiplication<\/strong><\/a> like the multiplicative identity (1). An identity matrix&#8217;s size is sometimes denoted by the subscript &#8220;n&#8221; to represent a n n matrix.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Characteristics_of_Identity_Matrix\"><\/span>Characteristics of Identity Matrix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>An Identity Matrix&#8217;s primary properties are:<\/strong><\/p>\n<p>An identity matrix is always a square matrix, meaning it contains an equal number of rows and columns.<\/p>\n<ul>\n<li><strong>Diagonal Elements:<\/strong> In an identity matrix, the diagonal elements are always equal to 1.<\/li>\n<li><strong>Non-Diagonal Elements:<\/strong> In an identity matrix, all non-diagonal elements are always equal to 0.<\/li>\n<li><strong>Size:<\/strong> An identity matrix&#8217;s size is dictated by its order, which is indicated by &#8220;n.&#8221; An n n identity matrix is made up of &#8220;n&#8221; rows and &#8220;n&#8221; columns. The outcome of multiplying an identity matrix by any compatible matrix of sufficient size is always the original matrix. To put it another way, it serves as a multiplicative identity for matrix multiplication.<\/li>\n<li><strong>Symbol:<\/strong> An identity matrix is frequently designated as &#8220;I&#8221; or &#8220;In,&#8221; where the subscript &#8220;n&#8221; reflects the matrix&#8217;s order or size.<\/li>\n<li><strong>Inverse:<\/strong> For square matrices, the identity matrix acts as the multiplicative identity. In terms of matrix multiplication, it may be thought of as the inverse of itself.<\/li>\n<\/ul>\n<p>The identity matrix is used extensively in linear algebra, particularly in matrix operations, solving systems of linear equations, and calculating eigenvalues and eigenvectors.<\/p>\n<p>The collection of input values for which the Identity Matrix is defined is referred to as the domain.<\/p>\n<p>The Identity Matrix&#8217;s range is the collection of output values.<\/p>\n<p>In linear transformations, multiplying a vector by an identity matrix produces the same vector while retaining its direction and magnitude.<\/p>\n<p>Commutativity, associativity, and distributivity in matrix operations are some of the features of the identity matrix.<\/p>\n<p>Remember that these qualities are unique to identity matrices and set them apart from other types of matrices.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Types_of_Identity_Matrices\"><\/span>Types of Identity Matrices<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The Identity matrix of order<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-667704\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Identity-matri.png\" alt=\"Identity matrix\" width=\"204\" height=\"87\" \/><\/p>\n<p>The Identity matrix of order 3<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-667705\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Identity-matrix-of-order-3-.png\" alt=\"Identity matrix of order 3\" width=\"213\" height=\"135\" \/><\/p>\n<p>The Identity matrix of order n: It is a square matrix having n rows and n columns, in which a<sub>ij<\/sub> = 0 for all and a<sub>ij<\/sub> = 1 for all i = j<\/p>\n<p><strong>Also Check For Relevant Topics:<\/strong><\/p>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/articles\/math-articles\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Articles<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/formulas\/math-formulas\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Math Formulas<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/symmetric-matrix\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Symmetric Matrix<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/topics\/symmetric-and-skew-symmetric-matrix\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 50px;\" type=\"button\">Symmetric and skew-symmetric matrix<\/button><\/a><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Steps_to_check_the_given_Matrix_is_Identity_Matrix_or_not\"><\/span>Steps to check the given Matrix is Identity Matrix or not<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>To determine whether a given matrix is an identity matrix, perform the following steps:<\/p>\n<ul>\n<li><strong>Check if the matrix is square:<\/strong> An identity matrix must be square, which means that the number of rows and columns must be equal. The matrix cannot be an identity matrix if it is not square.<\/li>\n<li><strong>Examine the diagonal elements:<\/strong> In an identity matrix, all diagonal elements (from top to bottom) are equal to 1. Iterate over the diagonal components until they are all 1. The matrix is not an identity matrix if any diagonal member is not equal to 1.<\/li>\n<li><strong>Check the non-diagonal elements:<\/strong> All non-diagonal elements (elements outside the main diagonal) in an identity matrix should be equal to 0. Iterate over the non-diagonal elements until they are all 0. The matrix is not an identity matrix if any non-diagonal member is not equal to 0.<\/li>\n<li>If the matrix meets both requirements 2 and 3, it is said to be an identity matrix.<\/li>\n<\/ul>\n<p>Examples for Identity Matrix.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-667706 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Examples-for-Identity-Matrix.png\" alt=\"Examples for Identity Matrix\" width=\"642\" height=\"211\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Examples-for-Identity-Matrix.png 642w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/08\/Examples-for-Identity-Matrix-300x99.png 300w\" sizes=\"(max-width: 642px) 100vw, 642px\" \/><\/p>\n<h2>Frequently asked questions on  (FAQs) Identity Matrix<\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Can_Identity_matrix_be_zero_matrix\"><\/span>Can Identity matrix be zero matrix? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tNo, any identity matrix is nonzero matrix, its principal diagonal elements are equal to 1 and remaining all elements are equal to zero.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Is_the_Identity_matrix_equal_to_1\"><\/span>Is the Identity matrix equal to 1?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe identity matrix is not at all equal to 1.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_shape_of_an_Identity_matrix\"><\/span>What is the shape of an Identity matrix? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe shape of identity matrix is square. The number of rows and number of columns must be equal in the Identity matrix. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_identity_and_null_matrix\"><\/span>What is the identity and null matrix ? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\t<strong>Matrix of Identity:<\/strong><br \/>\nA square matrix with one on the main diagonal and zeros everywhere else.<br \/>\nFor a n x n matrix, this is denoted by I or In.<br \/>\nIn matrix multiplication, it functions similarly to the number 1 in scalar multiplication.<br \/>\nWhen multiplied by any matrix, the outcome is the same.<br \/>\nMultiplying with the identity matrix is the inverse of any matrix.<br \/>\n<strong>The Null Matrix:<\/strong><br \/>\nAll of the elements of a matrix are zero.<br \/>\nFor a m x n matrix, this is denoted by O or O m by n.<br \/>\nIt serves no special mathematical use in matrix multiplication.<br \/>\nWhen this matrix is multiplied by any non-zero matrix, the output is always the null matrix. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Is_there_any_3_x_2_Identity_matrix\"><\/span>Is there any 3 x 2 Identity matrix? <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tNo, there is no rectangular matrix as an identity matrix, because identity matrix is a square matrix. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Are_Identity_matrices_being_only_square_matrices\"><\/span>Are Identity matrices being only square matrices <span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tYes, all Identity matrices are square matrices only \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Can Identity matrix be zero matrix? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"No, any identity matrix is nonzero matrix, its principal diagonal elements are equal to 1 and remaining all elements are equal to zero.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Is the Identity matrix equal to 1?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The identity matrix is not at all equal to 1.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the shape of an Identity matrix? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The shape of identity matrix is square. The number of rows and number of columns must be equal in the Identity matrix.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the identity and null matrix ? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"<strong>Matrix of Identity:<\/strong><br\/>\nA square matrix with one on the main diagonal and zeros everywhere else.<br\/>\nFor a n x n matrix, this is denoted by I or In.<br\/>\nIn matrix multiplication, it functions similarly to the number 1 in scalar multiplication.<br\/>\nWhen multiplied by any matrix, the outcome is the same.<br\/>\nMultiplying with the identity matrix is the inverse of any matrix.<br\/><strong>The Null Matrix:<\/strong><br\/>\nAll of the elements of a matrix are zero.<br\/>\nFor a m x n matrix, this is denoted by O or O m by n.<br\/>\nIt serves no special mathematical use in matrix multiplication.<br\/>\nWhen this matrix is multiplied by any non-zero matrix, the output is always the null matrix.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Is there any 3 x 2 Identity matrix? \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"No, there is no rectangular matrix as an identity matrix, because identity matrix is a square matrix.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Are Identity matrices being only square matrices \",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Yes, all Identity matrices are square matrices only\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>What is Identity Matrix? An identity matrix (I or In) is a square matrix with ones on its major diagonal [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Identity Matrix","_yoast_wpseo_title":"Identity Matrix - Definition, Properties and Types of Identity Matrices","_yoast_wpseo_metadesc":"An Identity Matrix is a square matrix where all diagonal elements are 1, and others are 0. 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