{"id":64894,"date":"2025-06-20T10:16:38","date_gmt":"2025-06-20T07:16:38","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64894"},"modified":"2025-06-20T10:16:38","modified_gmt":"2025-06-20T07:16:38","slug":"rank-3x3-matrix","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/rank-3x3-matrix\/","title":{"rendered":"Rank of 3\u00d73 Matrix 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 Rank Solver<\/h2><\/div><div class=\"relative group inline-block\">\n  <button class=\"favorite\" id=\"favorite\" data-favorite-id=\"64894\" data-favorite-title=\"Rank of 3\u00d73 Matrix\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/rank-3x3-matrix\/\" data-favorite-excerpt=\"This calculator determines the rank of a 3\u00d73 matrix using row reduction (Gaussian elimination) or determinant methods. It is useful for students, engineers, and professionals working with linear algebra and solving matrix equations.\" 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\/rank-3x3-matrix\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64894\" data-post=\"64894\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">a\u2081\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-1\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">a\u2081\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-2\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">a\u2081\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2082\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">a\u2082\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">a\u2082\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-7\"><label for=\"field-7\">a\u2083\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-8\"><label for=\"field-8\">a\u2083\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-9\"><label for=\"field-9\">a\u2083\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64894\">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>Matrix Rank<\/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><\/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=\"64894\" 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\">0<\/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\">0<\/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>Rank 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$$\n\\text{rank}(A) =\n\\begin{cases}\n3, &#038; \\det(A) \\neq 0 \\\\\n2, &#038; \\det(A) = 0 \\text{ and at least one 2\u00d72 minor} \\neq 0 \\\\\n1, &#038; \\text{all 2\u00d72 minors} = 0 \\text{ and at least one element} \\neq 0 \\\\\n0, &#038; A = 0\n\\end{cases}\n$$\n<\/div>\r\n\t<\/figure>\r\n\t<br \/>\n<strong>Where:<\/strong><\/p>\n<ul>\n<li>$$A$$ is a 3\u00d73 matrix<\/li>\n<li>$$\\det(A)$$ is the determinant of $$A$$<\/li>\n<li>Minors are determinants of 2\u00d72 submatrices<\/li>\n<\/ul>\n<p>\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>Rank of 3\u00d73 Matrix \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<p>\n$$A = \\begin{bmatrix} 1 &#038; 2 &#038; 3 \\\\ 4 &#038; 5 &#038; 6 \\\\ 7 &#038; 8 &#038; 9 \\end{bmatrix}$$<\/p>\n<p>$$\\det(A) = 1(5 \\cdot 9 &#8211; 6 \\cdot 8) &#8211; 2(4 \\cdot 9 &#8211; 6 \\cdot 7) + 3(4 \\cdot 8 &#8211; 5 \\cdot 7) = 1(-3) &#8211; 2(-6) + 3(-3) = -3 + 12 &#8211; 9 = 0$$<\/p>\n<p>Since $$\\(\\det(A)=0\\)$, check 2\u00d72 minors:<br \/>\n$$\\det \\begin{bmatrix} 1 &#038; 2 \\\\ 4 &#038; 5 \\end{bmatrix} = 1 \\cdot 5 &#8211; 2 \\cdot 4 = 5 &#8211; 8 = -3 \\neq 0 $$<br \/>\nSo, rank(A) = 2.<\/p>\n\r\n<\/section>\n<p>The rank of a 3\u00d73 matrix shows the number of linearly independent rows or columns. It is widely used in solving linear systems, determining matrix invertibility, and understanding matrix properties. Our calculator automates the process, providing accurate rank determination for any 3\u00d73 matrix.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This calculator determines the rank of a 3\u00d73 matrix using row reduction (Gaussian elimination) or determinant methods. It is useful for students, engineers, and professionals working with linear algebra and solving matrix equations.<\/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-64894","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\/64894","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=64894"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64894\/revisions"}],"predecessor-version":[{"id":64896,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64894\/revisions\/64896"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64894"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64894"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64894"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}