{"id":67393,"date":"2026-10-21T10:14:32","date_gmt":"2026-10-21T07:14:32","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=67393"},"modified":"2026-10-21T10:14:32","modified_gmt":"2026-10-21T07:14:32","slug":"matrix-determinant-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/matrix-determinant-calculator\/","title":{"rendered":"Matrix Determinant 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\">Calculate the Determinant of a Square Matrix<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"67393\" data-favorite-title=\"Determinant\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-determinant-calculator\/\" data-favorite-excerpt=\"Compute the exact determinant of any square matrix from 2\u00d72 to 6\u00d76. Enter your entries as integers, decimals or fractions and get the exact result, a decimal value, singularity check and step-by-step solution.\" 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-determinant-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-67393\" data-post=\"67393\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" 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><option value=\"4\">4 \u00d7 4<\/option><option value=\"5\">5 \u00d7 5<\/option><option value=\"6\">6 \u00d7 6<\/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\" id=\"input-2\"><label for=\"field-2\">Fill the grid with<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"fill_the_grid_with\"><option value=\"custom\" selected>Use the values entered below<\/option><option value=\"random\">Random integer matrix<\/option><option value=\"identity\">Identity matrix<\/option><option value=\"zeros\">Clear (all zeros)<\/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-3\"><span class=\"font-semibold\">Row 1<\/span><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-4\" aria-describedby=\"calc-info-4\" placeholder=\"0\" step=\"any\" value=\"2\"\/><small id=\"calc-info-4\">Integers, decimals (e.g. 1.5) or fractions (e.g. 3\/4). Empty cells count as 0.<\/small><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">a\u2081\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-5\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">a\u2081\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-6\" placeholder=\"0\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-7\" hidden><label for=\"field-7\">a\u2081\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-7\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-8\" hidden><label for=\"field-8\">a\u2081\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-8\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-9\" hidden><label for=\"field-9\">a\u2081\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-9\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-10\"><span class=\"font-semibold\">Row 2<\/span><\/div><div class=\"field\" id=\"input-11\"><label for=\"field-11\">a\u2082\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-11\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-12\"><label for=\"field-12\">a\u2082\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-12\" placeholder=\"0\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-13\"><label for=\"field-13\">a\u2082\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-13\" placeholder=\"0\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-14\" hidden><label for=\"field-14\">a\u2082\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-14\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-15\" hidden><label for=\"field-15\">a\u2082\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-15\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-16\" hidden><label for=\"field-16\">a\u2082\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-16\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-17\"><span class=\"font-semibold\">Row 3<\/span><\/div><div class=\"field\" id=\"input-18\"><label for=\"field-18\">a\u2083\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-18\" placeholder=\"0\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-19\"><label for=\"field-19\">a\u2083\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-19\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-20\"><label for=\"field-20\">a\u2083\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-20\" placeholder=\"0\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-21\" hidden><label for=\"field-21\">a\u2083\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-21\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-22\" hidden><label for=\"field-22\">a\u2083\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-22\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-23\" hidden><label for=\"field-23\">a\u2083\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-23\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-24\" hidden><span class=\"font-semibold\">Row 4<\/span><\/div><div class=\"field\" id=\"input-25\" hidden><label for=\"field-25\">a\u2084\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-25\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-26\" hidden><label for=\"field-26\">a\u2084\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-26\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-27\" hidden><label for=\"field-27\">a\u2084\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-27\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-28\" hidden><label for=\"field-28\">a\u2084\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-28\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-29\" hidden><label for=\"field-29\">a\u2084\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-29\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-30\" hidden><label for=\"field-30\">a\u2084\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-30\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-31\" hidden><span class=\"font-semibold\">Row 5<\/span><\/div><div class=\"field\" id=\"input-32\" hidden><label for=\"field-32\">a\u2085\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-32\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-33\" hidden><label for=\"field-33\">a\u2085\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-33\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-34\" hidden><label for=\"field-34\">a\u2085\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-34\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-35\" hidden><label for=\"field-35\">a\u2085\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-35\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-36\" hidden><label for=\"field-36\">a\u2085\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-36\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-37\" hidden><label for=\"field-37\">a\u2085\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-37\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-38\" hidden><span class=\"font-semibold\">Row 6<\/span><\/div><div class=\"field\" id=\"input-39\" hidden><label for=\"field-39\">a\u2086\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-39\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-40\" hidden><label for=\"field-40\">a\u2086\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-40\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-41\" hidden><label for=\"field-41\">a\u2086\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-41\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-42\" hidden><label for=\"field-42\">a\u2086\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-42\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-43\" hidden><label for=\"field-43\">a\u2086\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-43\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-44\" hidden><label for=\"field-44\">a\u2086\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-44\" placeholder=\"0\" step=\"any\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"67393\">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> Determinant Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Determinant (exact)<\/span><span class=\"result-term\">det A<\/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 class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Determinant (decimal)<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/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 class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>Is the matrix singular?<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/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 class=\"field-result is-column\" id=\"output-4\"><span class=\"field-title\"><span>Solution steps<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/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=\"67393\" 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>Determinant 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\\det \\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix} = ad - bc \\\\\n\\det \\begin{bmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\ a_{21} &amp; a_{22} &amp; a_{23} \\\\ a_{31} &amp; a_{32} &amp; a_{33} \\end{bmatrix} = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \\\\\n\\det A = \\sum_{j=1}^{n} (-1)^{1+j}\\, a_{1j}\\, M_{1j}\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 in row \\(i\\), column \\(j\\) of the square matrix \\(A\\)<\/li>\n<li>\\(M_{1j}\\) \u2014 the <em>minor<\/em>: the determinant of the submatrix left after deleting row 1 and column \\(j\\)<\/li>\n<li>\\((-1)^{1+j}M_{1j}\\) \u2014 the <em>cofactor<\/em> of the entry \\(a_{1j}\\)<\/li>\n<\/ul>\n<p>The first line is the diagonal rule for 2\u00d72 matrices, the second is cofactor (Laplace) expansion along the first row for 3\u00d73 matrices, and the last line is the general expansion that works for any \\(n \\times n\\) 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>Determinant \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the determinant of the 3\u00d73 matrix<\/p>\n<p>\\[ A = \\begin{bmatrix} 2 &amp; 1 &amp; 3 \\\\ 1 &amp; 0 &amp; 2 \\\\ 4 &amp; 1 &amp; 3 \\end{bmatrix} \\]<\/p>\n<p>\\[ M_{11} = 0 \\cdot 3 - 2 \\cdot 1 = -2, \\quad M_{12} = 1 \\cdot 3 - 2 \\cdot 4 = -5, \\quad M_{13} = 1 \\cdot 1 - 0 \\cdot 4 = 1 \\]<\/p>\n<p>\\[ \\det A = 2(-2) - 1(-5) + 3(1) = -4 + 5 + 3 \\]<\/p>\n<p><strong>Result:<\/strong> \\(\\det A = 4\\). The determinant is nonzero, so the matrix is invertible. You can reproduce this by pressing calculate with the tool's default values.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Determinant Calculator<\/h2>\n<ol>\n<li>Open the <strong>Matrix size<\/strong> dropdown and choose 2 \u00d7 2, 3 \u00d7 3, 4 \u00d7 4, 5 \u00d7 5 or 6 \u00d7 6. Only the fields of the selected size are used.<\/li>\n<li>Use the <strong>Fill the grid with<\/strong> dropdown to choose how to supply the matrix: <em>Use the values entered below<\/em>, <em>Random integer matrix<\/em> (a new matrix is rolled each time you switch), <em>Identity matrix<\/em>, or <em>Clear (all zeros)<\/em>.<\/li>\n<li>Type your entries into the fields a<sub>11<\/sub>, a<sub>12<\/sub>, \u2026 Integers, decimals such as 1.5, and fractions such as 3\/4 are all accepted. Empty cells count as 0.<\/li>\n<li>Read the results: <strong>Determinant (exact)<\/strong> gives the answer as an exact fraction or integer, <strong>Determinant (decimal)<\/strong> gives its decimal form, <strong>Is the matrix singular?<\/strong> tells you whether the matrix is invertible, and <strong>Solution steps<\/strong> shows the full working.<\/li>\n<\/ol>\n<h2>What Is the Determinant of a Matrix?<\/h2>\n<p>The determinant is a single number that summarizes an important geometric and algebraic property of a square matrix. Geometrically, the absolute value of the determinant of an \\(n \\times n\\) matrix is the volume scaling factor of the linear transformation that the matrix represents: \\(|\\det A|\\) tells you how much the transformation multiplies lengths (2\u00d72), areas (2\u00d72 and 3\u00d73 in 2D\/3D contexts) or volumes in general. A determinant of 3 means the transformation triples volume; a determinant of 0 means the transformation collapses space into a lower dimension and loses information.<\/p>\n<p>Algebraically, the determinant is a quick invertibility test. A square matrix has an inverse if and only if \\(\\det A \\neq 0\\). When \\(\\det A = 0\\) the matrix is called <em>singular<\/em>: its rows (and columns) are linearly dependent, systems \\(Ax = b\\) have no unique solution, and \\(A^{-1}\\) does not exist. This is why the determinant appears everywhere from solving linear systems (Cramer's rule) to finding eigenvalues and computing cross products.<\/p>\n<p>How you interpret the number depends on context. The exact value matters for hand-checks and for theory; the decimal value is convenient for estimation. The <em>sign<\/em> carries meaning too: a negative determinant means the transformation flips orientation, like a mirror reflection. Note that determinants are only defined for square matrices \u2014 a non-square matrix simply has no determinant.<\/p>\n<h2>How the Calculator Computes It: Two Methods<\/h2>\n<p>For 2\u00d72 and 3\u00d73 matrices the calculator uses <strong>cofactor expansion<\/strong> along the first row and shows every minor and product, including a rule-of-Sarrus cross-check for 3\u00d73 matrices. For 4\u00d74 to 6\u00d76 matrices it switches to <strong>Gaussian elimination<\/strong> with exact rational arithmetic: adding a multiple of one row to another leaves the determinant unchanged, while every row swap flips its sign. The determinant is then the product of the pivots of the resulting upper triangular matrix, times \\((-1)\\) for each swap performed. If no pivot can be found in some column, the matrix is singular and the determinant is exactly 0.<\/p>\n<p>Because all arithmetic is done with integers and fractions (not floating-point decimals), the result is <em>exact<\/em> \u2014 there is no rounding error, even for awkward fractions. Decimal entries with up to 12 places are converted internally to exact fractions, so 0.1 is treated as exactly 1\/10 rather than as an approximate binary value.<\/p>\n<h2>Useful Properties and Limitations<\/h2>\n<ul>\n<li>\\(\\det(AB) = \\det A \\cdot \\det B\\), and \\(\\det(A^{-1}) = 1\/\\det A\\) for an invertible matrix.<\/li>\n<li>Swapping two rows (or columns) multiplies the determinant by \u22121; multiplying a row by a constant \\(k\\) multiplies the determinant by \\(k\\). A matrix with two identical rows, or a zero row, has determinant 0.<\/li>\n<li>\\(\\det A = \\det A^{T}\\), so row and column arguments are interchangeable. You can explore the transpose itself with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-transpose-calculator\/\">Matrix Transpose calculator<\/a>.<\/li>\n<li>The calculator accepts sizes 2\u00d72 to 6\u00d76 only, with entries up to 12 decimal places. For larger matrices, use eliminations with cofactor expansion, or software built for big linear systems.<\/li>\n<li>A quick shortcut worth remembering: \\(\\det(kA) = k^{n} \\det A\\) for an \\(n \\times n\\) matrix, because every row gets multiplied by \\(k\\). The <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-scalar-multiplication-calculator\/\">Matrix Scalar Multiplication calculator<\/a> lets you build such matrices directly.<\/li>\n<li>Since a nonzero determinant guarantees a unique solution of a linear system, you may also want to solve systems directly with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/elimination-method\/\">Elimination Method calculator<\/a>, or compare related invariants such as the sum of diagonal entries using the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-trace-calculator\/\">Matrix Trace calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Compute the exact determinant of any square matrix from 2\u00d72 to 6\u00d76. Enter your entries as integers, decimals or fractions and get the exact result, a decimal value, singularity check and step-by-step solution.<\/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-67393","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\/67393","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=67393"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67393\/revisions"}],"predecessor-version":[{"id":67394,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/67393\/revisions\/67394"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=67393"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=67393"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=67393"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}