{"id":66838,"date":"2026-10-13T11:58:50","date_gmt":"2026-10-13T08:58:50","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66838"},"modified":"2026-10-13T11:58:50","modified_gmt":"2026-10-13T08:58:50","slug":"matrix-transpose-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/matrix-transpose-calculator\/","title":{"rendered":"Matrix Transpose 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\">Transpose a Matrix by Swapping Rows and Columns<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66838\" data-favorite-title=\"Matrix Transpose\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-transpose-calculator\/\" data-favorite-excerpt=\"Enter the entries of a 2 \u00d7 2 or 3 \u00d7 3 matrix and this calculator instantly swaps its rows and columns to produce the transposed matrix A\u1d40, shown as a table.\" 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-transpose-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66838\" data-post=\"66838\"><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=\"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><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66838\">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> Transposed Matrix (A\u1d40)<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Transposed matrix A\u1d40<\/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=\"66838\" 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 Transpose 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\\[\n\\begin{aligned}\nA &= \\begin{bmatrix} a_{11} & a_{12} & \\cdots & a_{1n} \\\\ a_{21} & a_{22} & \\cdots & a_{2n} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{n1} & a_{n2} & \\cdots & a_{nn} \\end{bmatrix}, \\quad\nA^{T} = \\begin{bmatrix} a_{11} & a_{21} & \\cdots & a_{n1} \\\\ a_{12} & a_{22} & \\cdots & a_{n2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a_{1n} & a_{2n} & \\cdots & a_{nn} \\end{bmatrix} \\\\[8pt]\n&\\left( A^{T} \\right)_{ij} = a_{ji}\n\\end{aligned}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a_{ij}\\) \u2014 the entry of matrix \\(A\\) in row \\(i\\) and column \\(j\\)<\/li>\n<li>\\(A^{T}\\) \u2014 the transpose of \\(A\\), whose entry in row \\(i\\), column \\(j\\) equals the entry of \\(A\\) in row \\(j\\), column \\(i\\)<\/li>\n<li>\\(n\\) \u2014 the number of rows and columns (2 or 3 for a square matrix)<\/li>\n<\/ul>\n<p>The formula says that transposing a matrix means reflecting it over its main diagonal: every row of \\(A\\) becomes a column of \\(A^{T}\\), and every column becomes a row.<\/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 Transpose \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> transpose the 3 \u00d7 3 matrix with entries 1 through 9, arranged row by row.<\/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>Swap rows and columns: the first column \\((1, 4, 7)\\) becomes the first row, the second column \\((2, 5, 8)\\) becomes the second row, and the third column \\((3, 6, 9)\\) becomes the third row.<\/p>\n<p>\\[ A^{T} = \\begin{bmatrix} 1 &amp; 4 &amp; 7 \\\\ 2 &amp; 5 &amp; 8 \\\\ 3 &amp; 6 &amp; 9 \\end{bmatrix} \\]<\/p>\n<p><strong>Result:<\/strong> the transposed matrix is \\(A^{T} = \\begin{bmatrix} 1 &amp; 4 &amp; 7 \\\\ 2 &amp; 5 &amp; 8 \\\\ 3 &amp; 6 &amp; 9 \\end{bmatrix}\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>In the <strong>Matrix size<\/strong> dropdown, choose <strong>2 \u00d7 2<\/strong> or <strong>3 \u00d7 3<\/strong> \u2014 the entry fields adjust automatically.<\/li>\n<li>Fill in every visible field of <strong>Matrix A<\/strong> (a\u2081\u2081, a\u2081\u2082, and so on) with the entries of your matrix, going row by row. The subscript tells you the position: a\u2081\u2082 is row 1, column 2.<\/li>\n<li>Read the result under <strong>Transposed matrix A\u1d40<\/strong>: it appears as a table with rows and columns swapped.<\/li>\n<\/ol>\n<p>Leave all fields at their default values (a 3 \u00d7 3 matrix filled with 1 through 9) and the result will match the worked example above, so you can verify everything works as expected.<\/p>\n<h2>What is the transpose of a matrix?<\/h2>\n<p>The transpose of a matrix \\(A\\), written \\(A^{T}\\) (or sometimes \\(A'\\)), is obtained by turning every row into a column and every column into a row. Visually, it is a reflection of the matrix across its main diagonal \u2014 the diagonal running from the top-left corner to the bottom-right corner. Entries on the main diagonal stay exactly where they are; everything else moves to the mirrored position.<\/p>\n<p>Although transposing looks like a purely mechanical operation, it carries real meaning. In data analysis, a table of measurements with subjects in rows and variables in columns becomes, after transposing, a table with variables in rows and subjects in columns. In computer graphics, transposes appear when converting between row-vector and column-vector conventions. In statistics, the transpose is a building block of the covariance matrix and of least-squares regression through expressions like \\(A^{T}A\\).<\/p>\n<p>Interpreting the result is straightforward: compare the first row of the output with the first column of your input \u2014 they must be identical. If they are, the transpose is correct. For a square matrix, you can also check a useful property: the transpose of the transpose returns the original matrix, \\((A^{T})^{T} = A\\).<\/p>\n<p>Two special cases are worth recognizing. If \\(A^{T} = A\\), the matrix is called <strong>symmetric<\/strong> \u2014 it reads the same across its diagonal. If \\(A^{T} = -A\\), it is <strong>skew-symmetric<\/strong>, which forces every diagonal entry to be zero. Try entering a symmetric matrix (for example, 1, 2, 3 in the first row and 2, 4, 5 in the second) and check that the output equals the input.<\/p>\n<h2>Properties, assumptions and tips<\/h2>\n<ul>\n<li><strong>Square matrices only.<\/strong> This calculator handles 2 \u00d7 2 and 3 \u00d7 3 matrices. The transpose itself is defined for any rectangular matrix \u2014 a 2 \u00d7 3 matrix transposes to a 3 \u00d7 2 matrix \u2014 but the entry fields here assume a square shape.<\/li>\n<li><strong>Basic algebraic rules.<\/strong> Transposing respects addition, \\((A + B)^{T} = A^{T} + B^{T}\\), and reverses the order of multiplication, \\((AB)^{T} = B^{T}A^{T}\\). It also commutes with scalar multiplication: \\((cA)^{T} = cA^{T}\\).<\/li>\n<li><strong>Any real numbers allowed.<\/strong> You can enter positive, negative, decimal, or zero values. Results are rounded to two decimal places for display, so values with more decimals may appear slightly shortened.<\/li>\n<li><strong>Watch the sign convention.<\/strong> The transpose is not the same as the inverse, and it is not the <em>conjugate transpose<\/em> used for complex matrices (where signs of imaginary parts are also flipped \u2014 the matrix analogue of taking a <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-conjugate\/\">complex conjugate<\/a>).<\/li>\n<\/ul>\n<p>If you are exploring related topics, a <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/magic-square-generator\/\">magic square<\/a> is a fun symmetric-style matrix puzzle, and the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/elimination-method\/\">elimination method<\/a> for solving linear systems is another place where matrices and their entries are manipulated row by row.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enter the entries of a 2 \u00d7 2 or 3 \u00d7 3 matrix and this calculator instantly swaps its rows and columns to produce the transposed matrix A\u1d40, shown as a table.<\/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-66838","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\/66838","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=66838"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66838\/revisions"}],"predecessor-version":[{"id":67093,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66838\/revisions\/67093"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}