{"id":66845,"date":"2026-10-15T09:20:25","date_gmt":"2026-10-15T06:20:25","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66845"},"modified":"2026-10-15T09:20:25","modified_gmt":"2026-10-15T06:20:25","slug":"matrix-scalar-multiplication-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/matrix-scalar-multiplication-calculator\/","title":{"rendered":"Matrix Scalar Multiplication 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\">Multiply a Matrix by a Scalar (Constant)<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66845\" data-favorite-title=\"Matrix Scalar Multiplication\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-scalar-multiplication-calculator\/\" data-favorite-excerpt=\"Multiply a 2 \u00d7 2 or 3 \u00d7 3 matrix by a scalar (constant). Enter the scalar k and the entries of matrix A to get the resulting matrix kA 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-scalar-multiplication-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66845\" data-post=\"66845\"><fieldset class=\"fieldset-input is-3\"><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\" selected>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=\"field is-single has-term\" id=\"input-2\"><label for=\"field-2\">Scalar (k)<\/label><div class=\"term\">k<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"scalar_k\" id=\"field-2\" aria-describedby=\"calc-info-2\" step=\"any\" value=\"3\"\/><small id=\"calc-info-2\">The constant every entry is multiplied by<\/small><\/div><div class=\"is-single field\" id=\"input-3\"><span class=\"font-semibold\">Matrix A<\/span><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">a\u2081\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-6\" hidden><label for=\"field-6\">a\u2081\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-7\"><label for=\"field-7\">a\u2082\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-8\"><label for=\"field-8\">a\u2082\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-9\" hidden><label for=\"field-9\">a\u2082\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"6\"\/><\/div><div class=\"field\" id=\"input-10\" hidden><label for=\"field-10\">a\u2083\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-10\" step=\"any\" value=\"7\"\/><\/div><div class=\"field\" id=\"input-11\" hidden><label for=\"field-11\">a\u2083\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-11\" step=\"any\" value=\"8\"\/><\/div><div class=\"field\" id=\"input-12\" hidden><label for=\"field-12\">a\u2083\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-12\" step=\"any\" value=\"9\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66845\">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>Resulting Matrix (kA)<\/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=\"66845\" 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\" 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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 Scalar Multiplication 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\\[ kA = k \\begin{bmatrix} a_{11} &amp; a_{12} &amp; \\cdots &amp; a_{1n} \\\\ a_{21} &amp; a_{22} &amp; \\cdots &amp; a_{2n} \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ a_{n1} &amp; a_{n2} &amp; \\cdots &amp; a_{nn} \\end{bmatrix} = \\begin{bmatrix} k\\,a_{11} &amp; k\\,a_{12} &amp; \\cdots &amp; k\\,a_{1n} \\\\ k\\,a_{21} &amp; k\\,a_{22} &amp; \\cdots &amp; k\\,a_{2n} \\\\ \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\ k\\,a_{n1} &amp; k\\,a_{n2} &amp; \\cdots &amp; k\\,a_{nn} \\end{bmatrix} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(k\\) \u2014 the scalar (any real constant)<\/li>\n<li>\\(A\\) \u2014 the original \\(n \\times n\\) matrix<\/li>\n<li>\\(a_{ij}\\) \u2014 the entry in row \\(i\\), column \\(j\\) of \\(A\\)<\/li>\n<li>\\(kA\\) \u2014 the resulting matrix, where every entry is \\(k\\) times the corresponding entry of \\(A\\)<\/li>\n<\/ul>\n<p>Scalar multiplication simply scales every entry of the matrix by the same constant. The size and shape of the matrix never change \u2014 only the values do.<\/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 Scalar Multiplication \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> multiply the 3 \u00d7 3 matrix \\(A\\) below by the scalar \\(k = 3\\).<\/p>\n<p>\\[ A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 4 &amp; 5 &amp; 6 \\\\ 7 &amp; 8 &amp; 9 \\end{bmatrix} \\]<\/p>\n<p>Each entry is multiplied by 3:<\/p>\n<p>\\[ 3A = \\begin{bmatrix} 3 \\cdot 1 &amp; 3 \\cdot 2 &amp; 3 \\cdot 3 \\\\ 3 \\cdot 4 &amp; 3 \\cdot 5 &amp; 3 \\cdot 6 \\\\ 3 \\cdot 7 &amp; 3 \\cdot 8 &amp; 3 \\cdot 9 \\end{bmatrix} = \\begin{bmatrix} 3 &amp; 6 &amp; 9 \\\\ 12 &amp; 15 &amp; 18 \\\\ 21 &amp; 24 &amp; 27 \\end{bmatrix} \\]<\/p>\n<p><strong>Result:<\/strong> the resulting matrix is \\(\\begin{bmatrix} 3 &amp; 6 &amp; 9 \\\\ 12 &amp; 15 &amp; 18 \\\\ 21 &amp; 24 &amp; 27 \\end{bmatrix}\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Choose the <strong>Matrix size<\/strong> \u2014 either 2 \u00d7 2 or 3 \u00d7 3 \u2014 from the dropdown. When you pick 2 \u00d7 2, the fields for the third row and column are hidden automatically.<\/li>\n<li>Enter the constant in the <strong>Scalar (k)<\/strong> field. It can be positive, negative, zero, or a decimal.<\/li>\n<li>Fill in every visible entry of <strong>Matrix A<\/strong> (a\u2081\u2081, a\u2081\u2082, a\u2081\u2083, a\u2082\u2081, and so on). The subscript tells you the position: the first number is the row, the second is the column.<\/li>\n<li>Read the <strong>Resulting Matrix (kA)<\/strong> shown under \u201cYour Results\u201d. Each cell displays the corresponding entry of \\(A\\) multiplied by \\(k\\), rounded to two decimal places.<\/li>\n<\/ol>\n<h2>What is scalar multiplication of a matrix?<\/h2>\n<p>In linear algebra, a <strong>scalar<\/strong> is a single number, as opposed to a matrix, which is a rectangular array of numbers. Multiplying a matrix by a scalar means multiplying <em>every single entry<\/em> of that matrix by the same number. The result is a new matrix of exactly the same dimensions \u2014 scalar multiplication never changes the size of a matrix.<\/p>\n<p>This operation is different from <em>matrix multiplication<\/em>, where two matrices are multiplied together using row-by-column dot products. Scalar multiplication is much simpler: it is just repeated ordinary multiplication applied entry by entry. For example, if \\(k = 2\\) and one entry of \\(A\\) is 5, the matching entry of \\(kA\\) is \\(2 \\times 5 = 10\\).<\/p>\n<p>Geometrically, if a matrix represents a transformation of space, multiplying it by a scalar \\(k\\) stretches (or shrinks) that transformation by a factor of \\(k\\) in every direction. With \\(k &gt; 1\\) the result is magnified, with \\(0 &lt; k &lt; 1\\) it is shrunk, and with a negative \\(k\\) the result is flipped through the origin: every entry changes sign.<\/p>\n<p>Scalar multiplication appears constantly in applied work: scaling coefficient matrices in systems of equations (related to the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/elimination-method\/\">elimination method<\/a>), normalizing data tables, adjusting weights, and building linear combinations of matrices. Note that scalar multiplication also plays nicely with matrix addition: \\(k(A + B) = kA + kB\\).<\/p>\n<h2>Key properties<\/h2>\n<ul>\n<li><strong>Order of the result:<\/strong> \\(kA\\) always has the same dimensions as \\(A\\).<\/li>\n<li><strong>Commutativity with the scalar:<\/strong> \\(kA = Ak\\) \u2014 the scalar can be written on either side.<\/li>\n<li><strong>Associativity:<\/strong> \\(c(kA) = (ck)A\\) \u2014 multiplying by two scalars in a row is the same as multiplying by their product once.<\/li>\n<li><strong>Distributivity over scalars:<\/strong> \\((c + k)A = cA + kA\\).<\/li>\n<li><strong>Distributivity over matrices:<\/strong> \\(k(A + B) = kA + kB\\).<\/li>\n<li><strong>Multiplying by 1:<\/strong> \\(1 \\cdot A = A\\) \u2014 the identity scalar leaves the matrix unchanged.<\/li>\n<li><strong>Multiplying by 0:<\/strong> \\(0 \\cdot A\\) is the zero matrix \u2014 every entry becomes 0.<\/li>\n<li><strong>Negative scalars:<\/strong> \\((-1)A\\) flips the sign of every entry, which is exactly the matrix \\(-A\\).<\/li>\n<\/ul>\n<h2>Limitations and practical tips<\/h2>\n<ul>\n<li>This calculator handles <strong>square matrices only<\/strong> (2 \u00d7 2 or 3 \u00d7 3). For rectangular matrices you can still apply the same rule by hand: multiply every entry by \\(k\\).<\/li>\n<li>Results are rounded to two decimal places for display, so entries with many decimals may appear slightly rounded. Use exact inputs (integers or simple fractions) when you need an exact answer.<\/li>\n<li>Very large scalars or entries can produce large results; the calculator works with floating-point numbers, so extreme values may lose precision.<\/li>\n<li>Remember the difference between \\(kA\\) (scalar multiplication) and \\(k \\cdot I\\) (a scalar times the identity matrix). They are not the same thing: \\(kI\\) has \\(k\\) on the diagonal and zeros elsewhere.<\/li>\n<li>If you are scaling a matrix as part of solving a linear system, each row operation must be applied to the whole augmented matrix \u2014 check the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/elimination-method\/\">elimination method calculator<\/a> for that workflow, or explore patterns in square grids with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/magic-square-generator\/\">magic square generator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Multiply a 2 \u00d7 2 or 3 \u00d7 3 matrix by a scalar (constant). Enter the scalar k and the entries of matrix A to get the resulting matrix kA instantly.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[69],"class_list":["post-66845","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\/66845","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=66845"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66845\/revisions"}],"predecessor-version":[{"id":67118,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66845\/revisions\/67118"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66845"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66845"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66845"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}