{"id":64885,"date":"2025-06-20T09:51:49","date_gmt":"2025-06-20T06:51:49","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64885"},"modified":"2026-09-30T20:49:47","modified_gmt":"2026-09-30T17:49:47","slug":"matrix-subtraction","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/matrix-subtraction\/","title":{"rendered":"Subtraction of Two Matrices Calculator"},"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 Subtraction Solver<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"64885\" data-favorite-title=\"Subtraction of Two Matrices\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-subtraction\/\" data-favorite-excerpt=\"Subtract a 2\u00d72 or 3\u00d73 matrix B from matrix A element by element. Enter the values of both matrices and get the resulting matrix C = A \u2212 B instantly, with the formula and a worked example.\" 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-subtraction\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64885\" data-post=\"64885\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Matrix Size<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"matrix_size\"><option value=\"2\">2 \u00d7 2<\/option><option value=\"3\">3 \u00d7 3<\/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=\"is-single field\" id=\"input-2\"><span class=\"font-semibold\">Matrix A<\/span><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">a\u2081\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-5\" hidden><label for=\"field-5\">a\u2081\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">a\u2082\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-7\"><label for=\"field-7\">a\u2082\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-8\" hidden><label for=\"field-8\">a\u2082\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"6\"\/><\/div><div class=\"field\" id=\"input-9\" hidden><label for=\"field-9\">a\u2083\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"7\"\/><\/div><div class=\"field\" id=\"input-10\" hidden><label for=\"field-10\">a\u2083\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-10\" step=\"any\" value=\"8\"\/><\/div><div class=\"field\" id=\"input-11\" hidden><label for=\"field-11\">a\u2083\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-11\" step=\"any\" value=\"9\"\/><\/div><div class=\"is-single field\" id=\"input-12\"><span class=\"font-semibold\">Matrix B<\/span><\/div><div class=\"field\" id=\"input-13\"><label for=\"field-13\">b\u2081\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-13\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-14\"><label for=\"field-14\">b\u2081\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-14\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-15\" hidden><label for=\"field-15\">b\u2081\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-15\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-16\"><label for=\"field-16\">b\u2082\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-16\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-17\"><label for=\"field-17\">b\u2082\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-17\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-18\" hidden><label for=\"field-18\">b\u2082\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-18\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-19\" hidden><label for=\"field-19\">b\u2083\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-19\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-20\" hidden><label for=\"field-20\">b\u2083\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-20\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-21\" hidden><label for=\"field-21\">b\u2083\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-21\" step=\"any\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64885\">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>Result<\/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=\"64885\" 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\">0<\/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 Subtraction 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 - B = \\begin{bmatrix} a_{11} - b_{11} &amp; a_{12} - b_{12} &amp; \\cdots &amp; a_{1n} - b_{1n} \\\\ a_{21} - b_{21} &amp; a_{22} - b_{22} &amp; \\cdots &amp; a_{2n} - b_{2n} \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ a_{m1} - b_{m1} &amp; a_{m2} - b_{m2} &amp; \\cdots &amp; a_{mn} - b_{mn} \\end{bmatrix} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(A\\) and \\(B\\) \u2014 matrices of the same size \\(m \\times n\\) (here 2\u00d72 or 3\u00d73)<\/li>\n<li>\\(C\\) \u2014 the resulting matrix of the same size \\(m \\times n\\)<\/li>\n<li>\\(a_{ij}\\), \\(b_{ij}\\) \u2014 the elements of \\(A\\) and \\(B\\) in row \\(i\\), column \\(j\\)<\/li>\n<\/ul>\n<p>Each element of the result is found by subtracting the corresponding element of \\(B\\) from the corresponding element of \\(A\\): \\(c_{ij} = a_{ij} - b_{ij}\\).<\/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>Subtraction of Two Matrices \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> subtract the identity matrix \\(B\\) from a 3\u00d73 matrix \\(A\\) using the calculator's default values.<\/p>\n<p>\\[ A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 4 &amp; 5 &amp; 6 \\\\ 7 &amp; 8 &amp; 9 \\end{bmatrix}, \\quad B = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix} \\]<\/p>\n<p>\\[ C = A - B = \\begin{bmatrix} 1-1 &amp; 2-0 &amp; 3-0 \\\\ 4-0 &amp; 5-1 &amp; 6-0 \\\\ 7-0 &amp; 8-0 &amp; 9-1 \\end{bmatrix} = \\begin{bmatrix} 0 &amp; 2 &amp; 3 \\\\ 4 &amp; 4 &amp; 6 \\\\ 7 &amp; 8 &amp; 8 \\end{bmatrix} \\]<\/p>\n<p><strong>Result:<\/strong> \\( C = \\begin{bmatrix} 0 &amp; 2 &amp; 3 \\\\ 4 &amp; 4 &amp; 6 \\\\ 7 &amp; 8 &amp; 8 \\end{bmatrix} \\)<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Subtraction of Two Matrices Calculator<\/h2>\n<ol>\n<li>Select the <strong>Matrix Size<\/strong> \u2014 either 2 \u00d7 2 or 3 \u00d7 3. The input fields adjust automatically.<\/li>\n<li>Enter the elements of <strong>Matrix A<\/strong> in the fields \\(a_{11}, a_{12}, \\ldots\\). The first index is the row, the second is the column.<\/li>\n<li>Enter the elements of <strong>Matrix B<\/strong> in the fields \\(b_{11}, b_{12}, \\ldots\\), in the same positions.<\/li>\n<li>The <strong>Result<\/strong> appears instantly as a table showing the matrix \\(C = A - B\\).<\/li>\n<li>Change any value and the result updates automatically \u2014 no button to press.<\/li>\n<\/ol>\n<h2>What Is Matrix Subtraction?<\/h2>\n<p>Matrix subtraction is one of the basic operations of linear algebra. Two matrices can be subtracted only when they have the same dimensions: an \\(m \\times n\\) matrix can be subtracted from another \\(m \\times n\\) matrix, and the result is again an \\(m \\times n\\) matrix. The operation is performed element by element, so each entry of the answer depends only on the two entries in the same position of the original matrices.<\/p>\n<p>It is worth remembering that subtraction is <em>not<\/em> commutative: in general \\(A - B \\neq B - A\\). In fact, \\(B - A = -(A - B)\\), so reversing the order gives the negated result. If you subtract a matrix from itself, you get the zero matrix, in which every element is 0.<\/p>\n<p>Matrix subtraction appears wherever matrices model data or transformations: solving systems of linear equations, comparing datasets (for example, subtracting an expected image from an observed one), computing residuals in statistics, physics simulations, computer graphics, and machine learning. It is also the inverse of matrix addition \u2014 adding \\(B\\) back to \\(C = A - B\\) recovers \\(A\\).<\/p>\n<p>When interpreting the result, read it position by position: the element in row \\(i\\), column \\(j\\) of \\(C\\) tells you how much smaller (or larger, if negative) the entry of \\(B\\) is than the entry of \\(A\\) at that position. Positive and negative entries can be mixed freely in the same result.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Same dimensions required:<\/strong> this calculator works with square matrices of size 2\u00d72 and 3\u00d73. Subtracting a 2\u00d72 matrix from a 3\u00d73 matrix is mathematically undefined.<\/li>\n<li><strong>Order matters:<\/strong> the calculator computes \\(A - B\\), not \\(B - A\\). To get \\(B - A\\), swap the values between the two input blocks.<\/li>\n<li><strong>Decimals and negatives:<\/strong> fractional and negative values are accepted; results are shown rounded to two decimal places.<\/li>\n<li><strong>Watch the signs:<\/strong> a common manual mistake is subtracting a negative entry incorrectly \u2014 \\(a_{ij} - (-b_{ij}) = a_{ij} + b_{ij}\\).<\/li>\n<li><strong>Subtraction of determinants is not the same:<\/strong> in general \\(\\det(A - B) \\neq \\det(A) - \\det(B)\\). Be careful not to confuse matrix operations with operations on determinants.<\/li>\n<\/ul>\n<h2>Related Calculators<\/h2>\n<p>To add matrices instead of subtracting, use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-addition\/\">Addition of Two Matrices calculator<\/a>. For products of matrices, see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-multiplication\/\">Matrix Multiplication calculator<\/a>, and to square a single matrix, try <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-squared\/\">Matrix Squared<\/a>. If your next step involves inverting a 3\u00d73 matrix, the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/inverse-matrix-3x3\/\">Inverse Matrix 3\u00d73 calculator<\/a> will help.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Subtract a 2\u00d72 or 3\u00d73 matrix B from matrix A element by element. Enter the values of both matrices and get the resulting matrix C = A \u2212 B instantly, with the formula and a worked example.<\/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-64885","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\/64885","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=64885"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64885\/revisions"}],"predecessor-version":[{"id":66291,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64885\/revisions\/66291"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64885"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64885"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64885"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}