{"id":64907,"date":"2025-06-20T10:42:04","date_gmt":"2025-06-20T07:42:04","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64907"},"modified":"2026-10-26T09:39:28","modified_gmt":"2026-10-26T06:39:28","slug":"inverse-matrix-3x3","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/inverse-matrix-3x3\/","title":{"rendered":"Inverse Matrix Calculator \u2013 2\u00d72, 3\u00d73 and 4\u00d74"},"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\">3\u00d73 Matrix Inverse Solver<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"64907\" data-favorite-title=\"Inverse Matrix 3\u00d73\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/inverse-matrix-3x3\/\" data-favorite-excerpt=\"Enter the nine entries of a 3\u00d73 matrix to find its inverse, determinant, and adjugate. The calculator uses the adjugate (cofactor) method and tells you when the matrix is singular.\" 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\/inverse-matrix-3x3\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64907\" data-post=\"64907\"><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\u00d72<\/option><option value=\"3\" selected>3\u00d73<\/option><option value=\"4\">4\u00d74<\/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\" id=\"input-2\"><label for=\"field-2\">Show steps<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"show_steps\"><option value=\"adj\" selected>Adjugate method: minors, cofactors, adjugate<\/option><option value=\"gauss\">Gauss\u2013Jordan elimination [A | I]<\/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-3\"><label for=\"field-3\">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=\"text\" name=\"a\" id=\"field-3\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">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=\"text\" name=\"a\" id=\"field-4\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">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=\"text\" name=\"a\" id=\"field-5\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-6\" hidden><label for=\"field-6\">a\u2081\u2084<\/label><div class=\"term\">a14<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"a\" id=\"field-6\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">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=\"text\" name=\"a\" id=\"field-7\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-8\"><label for=\"field-8\">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=\"text\" name=\"a\" id=\"field-8\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-9\"><label for=\"field-9\">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=\"text\" name=\"a\" id=\"field-9\" step=\"any\" value=\"4\"\/><\/div><div class=\"field has-term\" id=\"input-10\" hidden><label for=\"field-10\">a\u2082\u2084<\/label><div class=\"term\">a24<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"a\" id=\"field-10\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-11\"><label for=\"field-11\">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=\"text\" name=\"a\" id=\"field-11\" step=\"any\" value=\"5\"\/><\/div><div class=\"field has-term\" id=\"input-12\"><label for=\"field-12\">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=\"text\" name=\"a\" id=\"field-12\" step=\"any\" value=\"6\"\/><\/div><div class=\"field has-term\" id=\"input-13\"><label for=\"field-13\">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=\"text\" name=\"a\" id=\"field-13\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-14\" hidden><label for=\"field-14\">a\u2083\u2084<\/label><div class=\"term\">a34<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"a\" id=\"field-14\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-15\" hidden><label for=\"field-15\">a\u2084\u2081<\/label><div class=\"term\">a41<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"a\" id=\"field-15\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-16\" hidden><label for=\"field-16\">a\u2084\u2082<\/label><div class=\"term\">a42<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"a\" id=\"field-16\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-17\" hidden><label for=\"field-17\">a\u2084\u2083<\/label><div class=\"term\">a43<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"a\" id=\"field-17\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-18\" hidden><label for=\"field-18\">a\u2084\u2084<\/label><div class=\"term\">a44<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"a\" id=\"field-18\" step=\"any\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64907\">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\" id=\"output-1\"><span class=\"field-title\"><span>Determinant |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>Adjugate matrix A*<\/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>Inverse matrix A\u207b\u00b9<\/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>Inverse matrix (decimals)<\/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 class=\"field-result is-column\" id=\"output-5\"><span class=\"field-title\"><span>Step-by-step solution<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/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-6\"><span class=\"field-title\"><span>Check: A \u00b7 A\u207b\u00b9 = I<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/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=\"64907\" 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\">1<\/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>Inverse Matrix 3\u00d73 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\\[ A^{-1} = \\frac{1}{\\det(A)} \\operatorname{adj}(A) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(A\\) \u2014 the 3\u00d73 matrix being inverted<\/li>\n<li>\\(\\det(A)\\) \u2014 the determinant of \\(A\\)<\/li>\n<li>\\(\\operatorname{adj}(A)\\) \u2014 the adjugate (also called adjoint) matrix, i.e. the transpose of the cofactor matrix<\/li>\n<\/ul>\n<p>The matrix \\(A\\) is invertible only when \\(\\det(A) \\neq 0\\). If the determinant is zero, no inverse exists and the matrix is called singular.<\/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>Inverse Matrix 3\u00d73 \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> invert the matrix \\( A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 0 &amp; 1 &amp; 4 \\\\ 5 &amp; 6 &amp; 0 \\end{bmatrix} \\) (the default values of the calculator).<\/p>\n<p>\\[ \\det(A) = 1(1 \\cdot 0 - 4 \\cdot 6) - 2(0 \\cdot 0 - 4 \\cdot 5) + 3(0 \\cdot 6 - 1 \\cdot 5) = -24 + 40 - 15 = 1 \\]<\/p>\n<p>Since \\(\\det(A) = 1 \\neq 0\\), the inverse exists. The cofactor matrix is<\/p>\n<p>\\[ C = \\begin{bmatrix} -24 &amp; 18 &amp; 5 \\\\ 20 &amp; -15 &amp; -4 \\\\ -5 &amp; 4 &amp; 1 \\end{bmatrix} \\]<\/p>\n<p>and because the cofactor matrix is already symmetric here, \\(\\operatorname{adj}(A) = C\\). Therefore<\/p>\n<p>\\[ A^{-1} = \\frac{1}{1}\\operatorname{adj}(A) = \\begin{bmatrix} -24 &amp; 18 &amp; 5 \\\\ 20 &amp; -15 &amp; -4 \\\\ -5 &amp; 4 &amp; 1 \\end{bmatrix} \\]<\/p>\n<p><strong>Result:<\/strong> \\( A^{-1} = \\begin{bmatrix} -24 &amp; 18 &amp; 5 \\\\ 20 &amp; -15 &amp; -4 \\\\ -5 &amp; 4 &amp; 1 \\end{bmatrix} \\). You can verify it: \\(A \\cdot A^{-1} = I\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Inverse Matrix 3\u00d73 Calculator<\/h2>\n<ol>\n<li>Enter the nine elements of your matrix into the fields a\u2081\u2081 through a\u2083\u2083. The subscript tells you the position: a\u2081\u2081 is row 1, column 1; a\u2082\u2083 is row 2, column 3, and so on.<\/li>\n<li>Check that you have filled every field \u2014 leave none empty.<\/li>\n<li>Read the results: the calculator shows the <strong>Determinant |A|<\/strong>, the <strong>Adjugate matrix A*<\/strong>, and the <strong>Inverse matrix A\u207b\u00b9<\/strong>.<\/li>\n<li>If the determinant is 0, the calculator reports that the matrix is not invertible and there is nothing more to compute.<\/li>\n<\/ol>\n<h2>What Is the Inverse of a Matrix?<\/h2>\n<p>The inverse of a square matrix \\(A\\) is the matrix \\(A^{-1}\\) that satisfies \\(A \\cdot A^{-1} = A^{-1} \\cdot A = I\\), where \\(I\\) is the identity matrix (ones on the diagonal, zeros everywhere else). Multiplying a matrix by its inverse is the matrix analogue of dividing a number by itself: it \"undoes\" the transformation or operation that \\(A\\) represents.<\/p>\n<p>For a 3\u00d73 matrix the standard way to compute the inverse by hand is the <strong>adjugate method<\/strong>: find the determinant, build the matrix of cofactors, transpose it to get the adjugate, and divide by the determinant. This is exactly what the calculator does \u2014 quickly and without sign errors, which are the most common mistake when working through cofactors manually.<\/p>\n<p>Matrix inverses matter because many problems reduce to solving \\(A\\mathbf{x} = \\mathbf{b}\\) for \\(\\mathbf{x}\\). When \\(A\\) is invertible, the solution is simply \\(\\mathbf{x} = A^{-1}\\mathbf{b}\\). Inverses appear in solving linear systems, reversing geometric transformations in computer graphics, analyzing electrical networks, and many numerical methods. If you want to multiply the result by another matrix afterwards, our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-multiplication\/\">Matrix Multiplication<\/a> calculator can do that step for you.<\/p>\n<h2>Interpreting the Result<\/h2>\n<p>The calculator returns three things in order. First the <strong>determinant<\/strong>: if it is 0, the matrix is singular and the process stops \u2014 no inverse exists. Second the <strong>adjugate matrix<\/strong>, the intermediate result before dividing by the determinant. Third the <strong>inverse matrix<\/strong>, in which every entry of the adjugate has been divided by the determinant (rounded to four decimal places).<\/p>\n<p>A useful sanity check is that \\(\\det(A^{-1}) = 1\/\\det(A)\\), and that multiplying \\(A\\) by \\(A^{-1}\\) gives the identity matrix. If your application involves eigenvalues or matrix rank, you can continue with our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-3x3\/\">Eigenvalues and Eigenvectors 3\u00d73<\/a> or <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/rank-3x3-matrix\/\">Rank of 3\u00d73 Matrix<\/a> calculators.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Square matrices only.<\/strong> Only square matrices can have inverses, and this calculator handles exactly the 3\u00d73 case.<\/li>\n<li><strong>Singularity.<\/strong> A zero determinant means the matrix maps space onto a lower-dimensional subspace \u2014 information is lost, and the operation cannot be reversed.<\/li>\n<li><strong>Rounding.<\/strong> The inverse entries are displayed rounded to four decimal places; the underlying values may be repeating decimals or fractions such as \\(1\/3\\).<\/li>\n<li><strong>Numerical caution.<\/strong> A very small but nonzero determinant means the matrix is nearly singular. The inverse then has very large entries and small input errors can strongly distort the result \u2014 a common issue in real-world applied problems.<\/li>\n<li><strong>Tip for checking.<\/strong> After computing an inverse by hand, multiply \\(A \\cdot A^{-1}\\) mentally for one row or column; you should get 1 on the diagonal and 0 elsewhere.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter the nine entries of a 3\u00d73 matrix to find its inverse, determinant, and adjugate. The calculator uses the adjugate (cofactor) method and tells you when the matrix is singular.<\/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-64907","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\/64907","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=64907"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64907\/revisions"}],"predecessor-version":[{"id":67542,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64907\/revisions\/67542"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64907"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64907"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64907"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}