{"id":64907,"date":"2025-06-20T10:42:04","date_gmt":"2025-06-20T07:42:04","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64907"},"modified":"2025-06-20T10:42:04","modified_gmt":"2025-06-20T07:42:04","slug":"inverse-matrix-3x3","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/inverse-matrix-3x3\/","title":{"rendered":"Inverse Matrix 3\u00d73 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 class=\"text-lg font-display font-bold\">3\u00d73 Matrix Inverse Solver<\/h2><\/div><div class=\"relative group inline-block\">\n  <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=\"This calculator finds the inverse of a 3\u00d73 matrix using the adjoint method or row operations. It is essential for students, engineers, and professionals solving matrix equations, linear systems, or transformations in linear algebra.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-binary\">\n    <span class=\"icon icon-shape-star-empty\"><\/span>\n  <\/button>\n  <div class=\"absolute right-full -translate-y-1\/2 top-1\/2 mr-2 w-max max-w-xs px-3 py-2 bg-gray-800 text-white text-xs rounded shadow-lg opacity-0 group-hover:opacity-100 transition-opacity duration-200 z-10 pointer-events-none\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\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 has-term\" id=\"input-1\"><label for=\"field-1\">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=\"number\" name=\"a\" id=\"field-1\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">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=\"number\" name=\"a\" id=\"field-2\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">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=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">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=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"0\"\/><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">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=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">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=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"4\"\/><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">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=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"5\"\/><\/div><div class=\"field has-term\" id=\"input-8\"><label for=\"field-8\">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=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"6\"\/><\/div><div class=\"field has-term\" id=\"input-9\"><label for=\"field-9\">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=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"0\"\/><\/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 class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/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 class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/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 class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><\/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\"><h2 class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/h2><div class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button 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\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button 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\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button 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\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button 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\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button 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\">\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><ins class=\"adsbygoogle\"\n     style=\"display:block; text-align:center; margin: 32px 0;\"\n     data-ad-layout=\"in-article\"\n     data-ad-format=\"fluid\"\n     data-ad-client=\"ca-pub-1721569815777345\"\n     data-ad-slot=\"6317458308\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<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 of 3\u00d73 Matrix 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$$\nA^{-1} = \\frac{1}{\\det(A)} \\operatorname{adj}(A)\n$$\n<\/div>\r\n\t<\/figure>\r\n\t<br \/>\n<strong>Where:<\/strong><\/p>\n<ul>\n<li>$$A$$ is the 3\u00d73 matrix<\/li>\n<li>$$\\det(A)$$ is the determinant of $$A$$<\/li>\n<li>$$\\operatorname{adj}(A)$$ is the adjugate (transpose of cofactor matrix) of $$A$$<\/li>\n<\/ul>\n<p>The matrix $$A$$ is invertible only if $$\\det(A) \\neq 0$$.<br \/>\r\n<\/section>\n<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$$A = \\begin{bmatrix} 1 &#038; 2 &#038; 3 \\\\ 0 &#038; 1 &#038; 4 \\\\ 5 &#038; 6 &#038; 0 \\end{bmatrix}$$<\/p>\n<p>$$\\det(A) = 1(1 \\cdot 0 &#8211; 4 \\cdot 6) &#8211; 2(0 \\cdot 0 &#8211; 4 \\cdot 5) + 3(0 \\cdot 6 &#8211; 1 \\cdot 5) = 1(0 &#8211; 24) &#8211; 2(0 &#8211; 20) + 3(0 &#8211; 5) = -24 + 40 &#8211; 15 = 1$$<\/p>\n<p>Since \\(\\det(A) = 1\\), the inverse exists and:<br \/>\n$$A^{-1} = \\operatorname{adj}(A)$$<\/p>\n\r\n<\/section>\n<p>The inverse of a 3\u00d73 matrix is used in solving systems of linear equations, finding transformations, and applications in physics and engineering. Our calculator automates determinant, cofactor, and adjugate computations for accuracy and speed.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This calculator finds the inverse of a 3\u00d73 matrix using the adjoint method or row operations. It is essential for students, engineers, and professionals solving matrix equations, linear systems, or transformations in linear algebra.<\/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":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64907\/revisions"}],"predecessor-version":[{"id":64909,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64907\/revisions\/64909"}],"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}]}}