{"id":64897,"date":"2025-06-20T10:24:46","date_gmt":"2025-06-20T07:24:46","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64897"},"modified":"2026-10-26T09:35:20","modified_gmt":"2026-10-26T06:35:20","slug":"matrix-squared","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/matrix-squared\/","title":{"rendered":"Matrix Squared Calculator \u2013 A\u00b2 for 2\u00d72 and 3\u00d73"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Matrix Square Solver<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"64897\" data-favorite-title=\"Matrix Squared\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-squared\/\" data-favorite-excerpt=\"This calculator computes the square of a 2\u00d72 or 3\u00d73 matrix, i.e. the product A \u00d7 A. Enter the matrix elements and get the resulting squared matrix instantly.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-squared\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64897\" data-post=\"64897\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Select matrix size<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"select_matrix_size\"><option value=\"1\">2\u00d72<\/option><option value=\"2\">3\u00d73<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">a\u2081\u2081<\/label><div class=\"term\">a11<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-2\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">a\u2081\u2082<\/label><div class=\"term\">a12<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-4\" hidden><label for=\"field-4\">a\u2081\u2083<\/label><div class=\"term\">a13<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">a\u2082\u2081<\/label><div class=\"term\">a21<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">a\u2082\u2082<\/label><div class=\"term\">a22<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"4\"\/><\/div><div class=\"field has-term\" id=\"input-7\" hidden><label for=\"field-7\">a\u2082\u2083<\/label><div class=\"term\">a23<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-8\" hidden><label for=\"field-8\">a\u2083\u2081<\/label><div class=\"term\">a31<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-9\" hidden><label for=\"field-9\">a\u2083\u2082<\/label><div class=\"term\">a32<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-10\" hidden><label for=\"field-10\">a\u2083\u2083<\/label><div class=\"term\">a33<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-10\" step=\"any\" value=\"3\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64897\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Matrix Squared<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"64897\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">1<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Matrix Squared Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ C = A^2 = A \\times A, \\qquad C_{ij} = \\sum_{k=1}^{n} A_{ik} A_{kj} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(A\\) \u2014 a square matrix (2\u00d72 or 3\u00d73) with elements \\(A_{ij}\\)<\/li>\n<li>\\(C = A^2\\) \u2014 the matrix squared, obtained by multiplying \\(A\\) by itself<\/li>\n<li>\\(C_{ij}\\) \u2014 the element in row \\(i\\), column \\(j\\) of the result<\/li>\n<li>\\(n\\) \u2014 the size of the matrix (2 or 3)<\/li>\n<\/ul>\n<p>Each element of the result is the dot product of a row of \\(A\\) with a column of \\(A\\), exactly as in ordinary matrix multiplication \u2014 just with both factors being the same matrix.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Matrix Squared \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> square the 2\u00d72 matrix<\/p>\n<p>\\[ A = \\begin{bmatrix} 2 &amp; 1 \\\\ 1 &amp; 4 \\end{bmatrix} \\]<\/p>\n<p>\\[ A^2 = A \\times A = \\begin{bmatrix} 2\\cdot 2 + 1\\cdot 1 &amp; 2\\cdot 1 + 1\\cdot 4 \\\\ 1\\cdot 2 + 4\\cdot 1 &amp; 1\\cdot 1 + 4\\cdot 4 \\end{bmatrix} = \\begin{bmatrix} 5 &amp; 6 \\\\ 6 &amp; 17 \\end{bmatrix} \\]<\/p>\n<p><strong>Result:<\/strong> \\( A^2 = \\begin{bmatrix} 5 &amp; 6 \\\\ 6 &amp; 17 \\end{bmatrix} \\)<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Select the matrix size in the <strong>Select matrix size<\/strong> field: <strong>2\u00d72<\/strong> or <strong>3\u00d73<\/strong>. Choosing 3\u00d73 reveals the extra input fields.<\/li>\n<p>Enter the matrix elements in the fields <strong>a\u2081\u2081<\/strong>, <strong>a\u2081\u2082<\/strong>, \u2026 according to the row and column of each element. For a 2\u00d72 matrix you fill in a\u2081\u2081, a\u2081\u2082, a\u2082\u2081, a\u2082\u2082; for a 3\u00d73 matrix all nine fields.<\/p>\n<li>The squared matrix appears immediately in the <strong>Matrix Squared<\/strong> result panel.<\/li>\n<\/ol>\n<h2>What Does It Mean to Square a Matrix?<\/h2>\n<p>Squaring a matrix means multiplying it by itself: \\(A^2 = A \\times A\\). This is <em>not<\/em> the same as squaring each element individually \u2014 that would be elementwise (Hadamard) squaring and gives a completely different result. Matrix squaring follows the standard row-by-column multiplication rule, so every element of \\(A^2\\) mixes values from different rows and columns of \\(A\\).<\/p>\n<p>Only <strong>square matrices<\/strong> (same number of rows and columns) can be squared. A rectangular matrix such as 2\u00d73 cannot be multiplied by itself, because the number of columns of the left factor must equal the number of rows of the right factor.<\/p>\n<p>Squaring appears in many applications: it represents applying the same linear transformation twice (useful in computer graphics, Markov chains and graph theory, where \\(A^2\\) counts paths of length 2 between nodes), it appears in matrix polynomials and recurrences such as \\(x_{k+1} = A x_k\\), and it is a building block for computing higher matrix powers \\(A^m\\) by repeated squaring.<\/p>\n<p>A useful check on your result: \\(\\det(A^2) = \\det(A)^2\\). For the example above, \\(\\det(A) = 2\\cdot4 - 1\\cdot1 = 7\\) and \\(\\det(A^2) = 5\\cdot17 - 6\\cdot6 = 49 = 7^2\\). If you need to verify or compute this, our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/determinant-multiplication\/\">Determinant Multiplication<\/a> calculator can help.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Square matrices only.<\/strong> The calculator accepts 2\u00d72 and 3\u00d73 matrices. It cannot square rectangular matrices.<\/li>\n<li><strong>Not elementwise.<\/strong> Remember that \\((A^2)_{ij} \\neq (A_{ij})^2\\) in general. If you need each element squared separately, that is a different operation.<\/li>\n<li><strong>Order does not matter here.<\/strong> For a matrix multiplied by itself, \\(A \\times A\\) is unambiguous. But note that in general matrix multiplication is not commutative \u2014 \\(A \\times B \\neq B \\times A\\). Use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-multiplication\/\">Matrix Multiplication<\/a> calculator for products of two different matrices.<\/li>\n<li><strong>Negative and decimal values<\/strong> are accepted; results are shown rounded to two decimal places.<\/li>\n<li><strong>Identity and diagonal shortcuts.<\/strong> If \\(A\\) is diagonal, \\(A^2\\) is diagonal with squared diagonal entries; if \\(A = I\\), then \\(A^2 = I\\). These are quick sanity checks for your input.<\/li>\n<li>Squaring relates to eigenvalues: each eigenvalue \\(\\lambda\\) of \\(A\\) becomes \\(\\lambda^2\\) in \\(A^2\\). You can explore this with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-2x2\/\">Eigenvalues and Eigenvectors 2\u00d72<\/a> and <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-3x3\/\">Eigenvalues and Eigenvectors 3\u00d73<\/a> calculators.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>This calculator computes the square of a 2\u00d72 or 3\u00d73 matrix, i.e. the product A \u00d7 A. Enter the matrix elements and get the resulting squared matrix instantly.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[69],"class_list":["post-64897","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-matrix"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64897","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=64897"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64897\/revisions"}],"predecessor-version":[{"id":67536,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64897\/revisions\/67536"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64897"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64897"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64897"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}