{"id":64903,"date":"2025-06-20T10:36:48","date_gmt":"2025-06-20T07:36:48","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64903"},"modified":"2025-06-20T10:37:23","modified_gmt":"2025-06-20T07:37:23","slug":"eigenvalues-eigenvectors-2x2","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-2x2\/","title":{"rendered":"Eigenvalues and Eigenvectors 2\u00d72 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\">2\u00d72 Matrix Eigenvalues and Eigenvectors Solver<\/h2><\/div><div class=\"relative group inline-block\">\n  <button class=\"favorite\" id=\"favorite\" data-favorite-id=\"64903\" data-favorite-title=\"Eigenvalues and Eigenvectors 2\u00d72\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-2x2\/\" data-favorite-excerpt=\"This calculator computes the eigenvalues and eigenvectors of a 2\u00d72 matrix. It is perfect for students, engineers, and scientists working with linear algebra, physics, and systems analysis where eigen decomposition is required.\" 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\/eigenvalues-eigenvectors-2x2\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64903\" data-post=\"64903\"><fieldset class=\"fieldset-input\"><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=\"4\"\/><small>Matrix element at row 1, column 1<\/small><\/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\"\/><small>Matrix element at row 1, column 2<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">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-3\" step=\"any\" value=\"1\"\/><small>Matrix element at row 2, column 1<\/small><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">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-4\" step=\"any\" value=\"3\"\/><small>Matrix element at row 2, column 2<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64903\">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>|A| \u2014 Determinant of the matrix<\/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\" id=\"output-2\"><span class=\"field-title\"><span>Trace A \u2014 Sum of diagonal elements<\/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>Degenerate matrix (A \u2212 c\u00d7I)<\/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 class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>|A \u2212 c\u00d7I| \u2014 Characteristic polynomial<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Eigenvalue \u03bb\u2081<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-6\"><span class=\"field-title\"><span>Eigenvalue \u03bb\u2082<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-7\"><span class=\"field-title\"><span>Eigenvector for \u03bb\u2081 (x\u2081, x\u2082)<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-8\"><span class=\"field-title\"><span>Eigenvector for \u03bb\u2082 (x\u2081, x\u2082)<\/span><\/span><span class=\"field-value\" id=\"result-8\"><\/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=\"64903\" 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>Eigenvalues and Eigenvectors Formula for 2\u00d72 Matrix<\/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$$\n\\det(A &#8211; \\lambda I) = 0 \\\\[1em]\n(A &#8211; \\lambda I) \\vec{v} = 0 \\\\[1em]\n\\det\n\\begin{bmatrix}\na &#8211; \\lambda &#038; b \\\\\nc &#038; d &#8211; \\lambda\n\\end{bmatrix}\n= 0\n$$\n<\/div>\r\n\t<\/figure>\r\n\t<br \/>\n<strong>Where:<\/strong><\/p>\n<ul>\n<li>$$A = \\begin{bmatrix} a &#038; b \\\\ c &#038; d \\end{bmatrix}$$ is the 2\u00d72 matrix<\/li>\n<li>$$\\lambda$$ are the eigenvalues<\/li>\n<li>$$\\vec{v}$$ are the corresponding eigenvectors<\/li>\n<\/ul>\n<p>The characteristic equation is solved for $$\\lambda$$, then eigenvectors are found by solving the system $$(A &#8211; \\lambda I)\\vec{v} = 0$$.<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>Eigenvalues and Eigenvectors 2\u00d72 \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<p>\n$$A = \\begin{bmatrix} 4 &#038; 2 \\\\ 1 &#038; 3 \\end{bmatrix} $$<\/p>\n<p>$$\\det\\begin{bmatrix} 4 &#8211; \\lambda &#038; 2 \\\\ 1 &#038; 3 &#8211; \\lambda \\end{bmatrix} = (4 &#8211; \\lambda)(3 &#8211; \\lambda) &#8211; 2 \\cdot 1 = \\lambda^2 &#8211; 7 \\lambda + 10 = 0 $$<\/p>\n<p>Solutions (eigenvalues):<br \/>\n$$\\lambda_1 = 5, \\quad \\lambda_2 = 2$$<\/p>\n<p>Find eigenvectors by solving:<br \/>\n$$(A &#8211; \\lambda I)\\vec{v} = 0$$<\/p>\n<p>for each $$\\lambda$$.<\/p>\n\r\n<\/section>\n<p>Eigenvalues and eigenvectors of a 2\u00d72 matrix are essential in solving systems of equations, stability analysis, and understanding linear transformations. Our calculator helps find them quickly and accurately, avoiding manual calculation errors.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This calculator computes the eigenvalues and eigenvectors of a 2\u00d72 matrix. It is perfect for students, engineers, and scientists working with linear algebra, physics, and systems analysis where eigen decomposition is required.<\/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-64903","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\/64903","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=64903"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64903\/revisions"}],"predecessor-version":[{"id":64906,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64903\/revisions\/64906"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64903"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64903"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64903"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}