{"id":64888,"date":"2025-06-20T09:56:33","date_gmt":"2025-06-20T06:56:33","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64888"},"modified":"2026-09-30T20:50:45","modified_gmt":"2026-09-30T17:50:45","slug":"sum-of-determinants","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/sum-of-determinants\/","title":{"rendered":"Sum of Determinants Calculator"},"content":{"rendered":"<p><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\">Determinant Sum Solver<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"64888\" data-favorite-title=\"Sum of Determinants\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-of-determinants\/\" data-favorite-excerpt=\"Enter the entries of two square matrices (2 \u00d7 2 or 3 \u00d7 3) and this calculator finds each determinant and their sum |A| + |B| instantly, with step-by-step formulas explained.\" 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\/sum-of-determinants\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64888\" data-post=\"64888\"><fieldset class=\"fieldset-input\"><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 \u00d7 2<\/option><option value=\"3\">3 \u00d7 3<\/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-2\"><span class=\"font-semibold\">Matrix A<\/span><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">a\u2081\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-5\" hidden><label for=\"field-5\">a\u2081\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">a\u2082\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-7\"><label for=\"field-7\">a\u2082\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"5\"\/><\/div><div class=\"field\" id=\"input-8\" hidden><label for=\"field-8\">a\u2082\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"6\"\/><\/div><div class=\"field\" id=\"input-9\" hidden><label for=\"field-9\">a\u2083\u2081<\/label><input type=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"7\"\/><\/div><div class=\"field\" id=\"input-10\" hidden><label for=\"field-10\">a\u2083\u2082<\/label><input type=\"number\" name=\"a\" id=\"field-10\" step=\"any\" value=\"8\"\/><\/div><div class=\"field\" id=\"input-11\" hidden><label for=\"field-11\">a\u2083\u2083<\/label><input type=\"number\" name=\"a\" id=\"field-11\" step=\"any\" value=\"9\"\/><\/div><div class=\"is-single field\" id=\"input-12\"><span class=\"font-semibold\">Matrix B<\/span><\/div><div class=\"field\" id=\"input-13\"><label for=\"field-13\">b\u2081\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-13\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-14\"><label for=\"field-14\">b\u2081\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-14\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-15\" hidden><label for=\"field-15\">b\u2081\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-15\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-16\"><label for=\"field-16\">b\u2082\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-16\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-17\"><label for=\"field-17\">b\u2082\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-17\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-18\" hidden><label for=\"field-18\">b\u2082\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-18\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-19\" hidden><label for=\"field-19\">b\u2083\u2081<\/label><input type=\"number\" name=\"b\" id=\"field-19\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-20\" hidden><label for=\"field-20\">b\u2083\u2082<\/label><input type=\"number\" name=\"b\" id=\"field-20\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-21\" hidden><label for=\"field-21\">b\u2083\u2083<\/label><input type=\"number\" name=\"b\" id=\"field-21\" step=\"any\" value=\"1\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64888\">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|<\/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\" id=\"output-2\"><span class=\"field-title\"><span>|B|<\/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\" id=\"output-3\"><span class=\"field-title\"><span>|A| + |B|<\/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><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=\"64888\" 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><br \/>\n<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\">Sum of Determinants \u2013 Calculation Example<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"64888\" data-favorite-title=\"Sum of Determinants\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-of-determinants\/\" data-favorite-excerpt=\"Enter the entries of two square matrices (2 \u00d7 2 or 3 \u00d7 3) and this calculator finds each determinant and their sum |A| + |B| instantly, with step-by-step formulas explained.\" 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\/sum-of-determinants\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64888\" data-post=\"64888\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">First loan amount<\/label><div class=\"term\">L1<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"first_loan_amount\" id=\"field-1\" aria-describedby=\"calc-unit-1 calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"100000\"\/><div class=\"unit\" id=\"calc-unit-1\">$<\/div><small id=\"calc-info-1\">Enter the amount of the first loan<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">First loan interest rate<\/label><div class=\"term\">r1<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"first_loan_interest_rate\" id=\"field-2\" aria-describedby=\"calc-unit-2 calc-info-2\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" max=\"100\" value=\"4\"\/><div class=\"unit\" id=\"calc-unit-2\">%<\/div><small id=\"calc-info-2\">Enter the interest rate of the first loan<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Second loan amount<\/label><div class=\"term\">L2<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"second_loan_amount\" id=\"field-3\" aria-describedby=\"calc-unit-3 calc-info-3\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" value=\"50000\"\/><div class=\"unit\" id=\"calc-unit-3\">$<\/div><small id=\"calc-info-3\">Enter the amount of the second loan<\/small><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">Second loan interest rate<\/label><div class=\"term\">r2<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"text\" name=\"second_loan_interest_rate\" id=\"field-4\" aria-describedby=\"calc-unit-4 calc-info-4\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" min=\"0\" max=\"100\" value=\"6\"\/><div class=\"unit\" id=\"calc-unit-4\">%<\/div><small id=\"calc-info-4\">Enter the interest rate of the second loan<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64888\">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>Blended Interest Rate<\/span><\/span><span class=\"field-value\" id=\"result-1\" data-unit=\"%\"><\/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=\"64888\" 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><\/p>\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>Sum of Determinants 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| + |B| = \\det(A) + \\det(B) \\]\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<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(A\\) and \\(B\\) \u2014 square matrices of the same size (2 \u00d7 2 or 3 \u00d7 3)<\/li>\n<li>\\(\\det(A)\\), written \\(|A|\\) \u2014 the determinant of matrix \\(A\\)<\/li>\n<li>\\(\\det(B)\\), written \\(|B|\\) \u2014 the determinant of matrix \\(B\\)<\/li>\n<\/ul>\n<p>The first line states the operation itself: each determinant is computed independently, and the two scalar results are then added together. The second and third lines show how a single determinant is evaluated for 2 \u00d7 2 and 3 \u00d7 3 matrices (cofactor expansion along the first row).<\/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>Sum of Determinants \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> two 3 \u00d7 3 matrices with the calculator's default values:<\/p>\n<p>\\[ A = \\begin{bmatrix} 1 &amp; 2 &amp; 3 \\\\ 4 &amp; 5 &amp; 6 \\\\ 7 &amp; 8 &amp; 9 \\end{bmatrix}, \\quad B = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix} \\]<\/p>\n<p>\\[ \\det(A) = 1(5 \\cdot 9 - 6 \\cdot 8) - 2(4 \\cdot 9 - 6 \\cdot 7) + 3(4 \\cdot 8 - 5 \\cdot 7) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0 \\]<\/p>\n<p>\\[ \\det(B) = 1(1 \\cdot 1 - 0 \\cdot 0) - 0(0 \\cdot 1 - 0 \\cdot 0) + 0(0 \\cdot 0 - 1 \\cdot 0) = 1 \\]<\/p>\n<p>\\[ |A| + |B| = 0 + 1 = 1 \\]<\/p>\n<p><strong>Result:<\/strong> \\(\\det(A) = 0\\), \\(\\det(B) = 1\\), and the sum of the determinants is \\(1\\). Note that matrix \\(A\\) is singular (its rows are linearly dependent), which is why its determinant is zero.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Select the <strong>Matrix Size<\/strong> \u2014 either 2 \u00d7 2 or 3 \u00d7 3 \u2014 from the dropdown at the top. The 3 \u00d7 3 fields appear or hide automatically.<\/li>\n<li>Enter the entries of <strong>Matrix A<\/strong> in the fields labeled a\u2081\u2081 through a\u2083\u2083 (row by row).<\/li>\n<li>Enter the entries of <strong>Matrix B<\/strong> in the fields labeled b\u2081\u2081 through b\u2083\u2083.<\/li>\n<li>The results update instantly: <strong>|A|<\/strong>, <strong>|B|<\/strong>, and the sum <strong>|A| + |B|<\/strong>.<\/li>\n<\/ol>\n<h2>What Is the Sum of Determinants?<\/h2>\n<p>A determinant is a single number computed from a square matrix. It encodes key properties of the linear transformation the matrix represents: whether the transformation is invertible, how it scales areas or volumes, and how it orients space. A determinant of zero means the matrix is singular \u2014 it collapses space and has no inverse.<\/p>\n<p>The sum of determinants is simply the scalar addition \\(\\det(A) + \\det(B)\\). Each matrix is evaluated on its own, and only the two resulting numbers are combined. This is a common exercise in linear algebra courses and appears in proofs and formulas where determinants of several matrices must be pooled \u2014 for example, when checking linear dependence, evaluating block-matrix expressions, or combining characteristic polynomials.<\/p>\n<p>A crucial point: <strong>the sum of determinants is generally not equal to the determinant of the sum<\/strong>. In symbols, \\(\\det(A + B) \\neq \\det(A) + \\det(B)\\) in almost all cases. The determinant is a nonlinear, multiplicative function \u2014 it satisfies \\(\\det(AB) = \\det(A)\\,\\det(B)\\), which you can verify with our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/determinant-multiplication\/\">Determinant Multiplication<\/a> calculator \u2014 but it does not distribute over matrix addition. If you need the determinant of \\(A + B\\), first add the matrices element by element with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-addition\/\">Matrix Addition<\/a> calculator, then compute the determinant of the result.<\/p>\n<p>Interpreting the output is straightforward: <strong>|A|<\/strong> and <strong>|B|<\/strong> are the two individual determinants, and <strong>|A| + |B|<\/strong> is their ordinary arithmetic sum, which can be positive, negative, or zero. A zero sum does not imply either matrix is singular \u2014 one determinant can simply cancel the other.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Same size, square matrices only.<\/strong> Determinants are defined only for square matrices, and this tool accepts 2 \u00d7 2 and 3 \u00d7 3 inputs. The two matrices do not need to be related in any other way.<\/li>\n<li><strong>Signs matter.<\/strong> The determinant can be negative. Swapping two rows or two columns flips the sign, so double-check the order in which you enter values (row first, then column: a\u2082\u2081 is row 2, column 1).<\/li>\n<li><strong>Round-off.<\/strong> The calculator rounds the displayed results, so very large entries or numbers with many decimals may lose a little precision in the shown output.<\/li>\n<li><strong>Singular matrices.<\/strong> If a determinant comes out as 0, the matrix has no inverse and its rows (and columns) are linearly dependent. To explore such a matrix further, try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/rank-3x3-matrix\/\">Rank of 3\u00d73 Matrix<\/a> calculator or the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/inverse-matrix-3x3\/\">Inverse Matrix 3\u00d73<\/a> calculator.<\/li>\n<li><strong>Need subtraction instead?<\/strong> For the difference \\(\\det(A) - \\det(B)\\), you can enter \\(-B\\) as the second matrix, since \\(\\det(-B) = -\\det(B)\\) for 3 \u00d7 3 (and, more generally, \\(\\det(-B) = (-1)^n\\det(B)\\) for an \\(n \\times n\\) matrix). Element-by-element matrix subtraction is covered by our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/matrix-subtraction\/\">Subtraction of Two Matrices<\/a> calculator.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter the entries of two square matrices (2 \u00d7 2 or 3 \u00d7 3) and this calculator finds each determinant and their sum |A| + |B| instantly, with step-by-step formulas explained.<\/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-64888","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\/64888","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=64888"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64888\/revisions"}],"predecessor-version":[{"id":66294,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64888\/revisions\/66294"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64888"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64888"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64888"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}