{"id":3013,"date":"2026-11-04T07:26:45","date_gmt":"2026-11-04T07:26:45","guid":{"rendered":"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-determinante-de-matriz\/"},"modified":"2026-11-04T07:26:45","modified_gmt":"2026-11-04T07:26:45","slug":"calculadora-de-determinante-de-matriz","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-determinante-de-matriz\/","title":{"rendered":"Calculadora de determinante de matriz"},"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\">Calcule o determinante de uma matriz quadrada<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"3013\" data-favorite-title=\"Determinante\" data-favorite-url=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-determinante-de-matriz\/\" data-favorite-excerpt=\"Calcule o determinante exato de qualquer matriz quadrada de 2\u00d72 a 6\u00d76. Digite os elementos como inteiros, decimais ou fra\u00e7\u00f5es e obtenha o resultado exato, o valor decimal, a verifica\u00e7\u00e3o de singularidade e a solu\u00e7\u00e3o passo a passo.\" aria-label=\"Adicionar aos favoritos\" 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\/pt\/matematica\/calculadora-de-determinante-de-matriz\/\" method=\"POST\" class=\"calculator\" id=\"calculator-3013\" data-post=\"3013\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Campos de entrada<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Ordem da matriz<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"ordem_da_matriz\"><option value=\"2\">2 \u00d7 2<\/option><option value=\"3\" selected>3 \u00d7 3<\/option><option value=\"4\">4 \u00d7 4<\/option><option value=\"5\">5 \u00d7 5<\/option><option value=\"6\">6 \u00d7 6<\/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\" id=\"input-2\"><label for=\"field-2\">Preencher a matriz com<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"preencher_a_matriz_com\"><option value=\"custom\" selected>Usar os valores digitados abaixo<\/option><option value=\"random\">Matriz aleat\u00f3ria de inteiros<\/option><option value=\"identity\">Matriz identidade<\/option><option value=\"zeros\">Zerar (todos os valores zero)<\/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-3\"><span class=\"font-semibold\">Linha 1<\/span><\/div><div class=\"field\" id=\"input-4\"><label for=\"field-4\">a\u2081\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-4\" aria-describedby=\"calc-info-4\" placeholder=\"0\" step=\"any\" value=\"2\"\/><small id=\"calc-info-4\">N\u00fameros inteiros, decimais (ex.: 1,5) ou fra\u00e7\u00f5es (ex.: 3\/4). C\u00e9lulas vazias contam como 0.<\/small><\/div><div class=\"field\" id=\"input-5\"><label for=\"field-5\">a\u2081\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-5\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-6\"><label for=\"field-6\">a\u2081\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-6\" placeholder=\"0\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-7\" hidden><label for=\"field-7\">a\u2081\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-7\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-8\" hidden><label for=\"field-8\">a\u2081\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-8\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-9\" hidden><label for=\"field-9\">a\u2081\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-9\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-10\"><span class=\"font-semibold\">Linha 2<\/span><\/div><div class=\"field\" id=\"input-11\"><label for=\"field-11\">a\u2082\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-11\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-12\"><label for=\"field-12\">a\u2082\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-12\" placeholder=\"0\" step=\"any\" value=\"0\"\/><\/div><div class=\"field\" id=\"input-13\"><label for=\"field-13\">a\u2082\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-13\" placeholder=\"0\" step=\"any\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-14\" hidden><label for=\"field-14\">a\u2082\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-14\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-15\" hidden><label for=\"field-15\">a\u2082\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-15\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-16\" hidden><label for=\"field-16\">a\u2082\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-16\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-17\"><span class=\"font-semibold\">Linha 3<\/span><\/div><div class=\"field\" id=\"input-18\"><label for=\"field-18\">a\u2083\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-18\" placeholder=\"0\" step=\"any\" value=\"4\"\/><\/div><div class=\"field\" id=\"input-19\"><label for=\"field-19\">a\u2083\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-19\" placeholder=\"0\" step=\"any\" value=\"1\"\/><\/div><div class=\"field\" id=\"input-20\"><label for=\"field-20\">a\u2083\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-20\" placeholder=\"0\" step=\"any\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-21\" hidden><label for=\"field-21\">a\u2083\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-21\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-22\" hidden><label for=\"field-22\">a\u2083\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-22\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-23\" hidden><label for=\"field-23\">a\u2083\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-23\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-24\" hidden><span class=\"font-semibold\">Linha 4<\/span><\/div><div class=\"field\" id=\"input-25\" hidden><label for=\"field-25\">a\u2084\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-25\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-26\" hidden><label for=\"field-26\">a\u2084\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-26\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-27\" hidden><label for=\"field-27\">a\u2084\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-27\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-28\" hidden><label for=\"field-28\">a\u2084\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-28\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-29\" hidden><label for=\"field-29\">a\u2084\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-29\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-30\" hidden><label for=\"field-30\">a\u2084\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-30\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-31\" hidden><span class=\"font-semibold\">Linha 5<\/span><\/div><div class=\"field\" id=\"input-32\" hidden><label for=\"field-32\">a\u2085\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-32\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-33\" hidden><label for=\"field-33\">a\u2085\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-33\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-34\" hidden><label for=\"field-34\">a\u2085\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-34\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-35\" hidden><label for=\"field-35\">a\u2085\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-35\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-36\" hidden><label for=\"field-36\">a\u2085\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-36\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-37\" hidden><label for=\"field-37\">a\u2085\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-37\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"is-single field\" id=\"input-38\" hidden><span class=\"font-semibold\">Linha 6<\/span><\/div><div class=\"field\" id=\"input-39\" hidden><label for=\"field-39\">a\u2086\u2081<\/label><input type=\"text\" name=\"a\" id=\"field-39\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-40\" hidden><label for=\"field-40\">a\u2086\u2082<\/label><input type=\"text\" name=\"a\" id=\"field-40\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-41\" hidden><label for=\"field-41\">a\u2086\u2083<\/label><input type=\"text\" name=\"a\" id=\"field-41\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-42\" hidden><label for=\"field-42\">a\u2086\u2084<\/label><input type=\"text\" name=\"a\" id=\"field-42\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-43\" hidden><label for=\"field-43\">a\u2086\u2085<\/label><input type=\"text\" name=\"a\" id=\"field-43\" placeholder=\"0\" step=\"any\"\/><\/div><div class=\"field\" id=\"input-44\" hidden><label for=\"field-44\">a\u2086\u2086<\/label><input type=\"text\" name=\"a\" id=\"field-44\" placeholder=\"0\" step=\"any\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalcular\" id=\"calculate-button\" data-post=\"3013\">Calcular<\/button><button type=\"reset\">Redefinir<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calcular automaticamente<\/label><small>Se ativado, o resultado \u00e9 atualizado automaticamente quando voc\u00ea altera qualquer valor.<\/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> Resultados do determinante<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Determinante (exato)<\/span><span class=\"result-term\">det A<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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>Determinante (decimal)<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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>A matriz \u00e9 singular?<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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>Passos da solu\u00e7\u00e3o<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copiar resultado\" aria-label=\"Copiar resultado\"><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\">Copiar link para este c\u00e1lculo<\/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>Deixar um coment\u00e1rio<\/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=\"3013\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">O que voc\u00ea achou deste resultado?<\/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\">Curti <\/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\">\u00datil <\/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\">Confuso <\/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\">Decepcionado <\/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\">Impreciso <\/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>F\u00f3rmula do determinante<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">F\u00f3rmula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[\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\\det A = \\sum_{j=1}^{n} (-1)^{1+j}\\, a_{1j}\\, M_{1j}\n\\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Onde:<\/strong><\/p>\n<ul>\n<li>\\(a_{ij}\\) \u2014 o elemento da linha \\(i\\) e coluna \\(j\\) da matriz quadrada \\(A\\)<\/li>\n<li>\\(M_{1j}\\) \u2014 o <em>menor complementar<\/em>: o determinante da submatriz que resta ap\u00f3s excluir a linha 1 e a coluna \\(j\\)<\/li>\n<li>\\((-1)^{1+j}M_{1j}\\) \u2014 o <em>cofator<\/em> do elemento \\(a_{1j}\\)<\/li>\n<\/ul>\n<p>A primeira linha \u00e9 a regra das diagonais para matrizes 2\u00d72, a segunda \u00e9 a expans\u00e3o em cofatores (Laplace) ao longo da primeira linha para matrizes 3\u00d73, e a \u00faltima \u00e9 a expans\u00e3o geral, que vale para qualquer matriz \\(n \\times n\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.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>Determinante \u2013 Exemplo de c\u00e1lculo<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemplo:<\/strong> encontre o determinante da matriz 3\u00d73<\/p>\n<p>\\[ A = \\begin{bmatrix} 2 &amp; 1 &amp; 3 \\\\ 1 &amp; 0 &amp; 2 \\\\ 4 &amp; 1 &amp; 3 \\end{bmatrix} \\]<\/p>\n<p>\\[ M_{11} = 0 \\cdot 3 - 2 \\cdot 1 = -2, \\quad M_{12} = 1 \\cdot 3 - 2 \\cdot 4 = -5, \\quad M_{13} = 1 \\cdot 1 - 0 \\cdot 4 = 1 \\]<\/p>\n<p>\\[ \\det A = 2(-2) - 1(-5) + 3(1) = -4 + 5 + 3 \\]<\/p>\n<p><strong>Resultado:<\/strong> \\(\\det A = 4\\). Como o determinante \u00e9 diferente de zero, a matriz \u00e9 invert\u00edvel. Voc\u00ea pode reproduzir esse resultado clicando em Calcular com os valores padr\u00e3o da ferramenta.<\/p>\n<p>\n\r\n<\/section>\n<h2>Como usar a calculadora de determinante<\/h2>\n<ol>\n<li>Abra o menu suspenso <strong>Tamanho da matriz<\/strong> e escolha 2 \u00d7 2, 3 \u00d7 3, 4 \u00d7 4, 5 \u00d7 5 ou 6 \u00d7 6. Apenas os campos do tamanho selecionado s\u00e3o usados.<\/li>\n<li>Use o menu suspenso <strong>Preencher a grade com<\/strong> para escolher como informar a matriz: <em>Usar os valores digitados abaixo<\/em>, <em>Matriz aleat\u00f3ria de inteiros<\/em> (uma nova matriz \u00e9 sorteada a cada troca de op\u00e7\u00e3o), <em>Matriz identidade<\/em> ou <em>Limpar (todos os zeros)<\/em>.<\/li>\n<li>Digite os elementos nos campos a<sub>11<\/sub>, a<sub>12<\/sub>, \u2026 Inteiros, decimais como 1,5 e fra\u00e7\u00f5es como 3\/4 s\u00e3o todos aceitos. C\u00e9lulas vazias contam como 0.<\/li>\n<li>Confira os resultados: <strong>Determinante (exato)<\/strong> traz a resposta como uma fra\u00e7\u00e3o exata ou um n\u00famero inteiro, <strong>Determinante (decimal)<\/strong> mostra a forma decimal, <strong>A matriz \u00e9 singular?<\/strong> indica se a matriz \u00e9 invert\u00edvel, e <strong>Etapas da solu\u00e7\u00e3o<\/strong> mostra todo o desenvolvimento.<\/li>\n<\/ol>\n<h2>O que \u00e9 o determinante de uma matriz?<\/h2>\n<p>O determinante \u00e9 um \u00fanico n\u00famero que resume uma importante propriedade geom\u00e9trica e alg\u00e9brica de uma matriz quadrada. Geometricamente, o valor absoluto do determinante de uma matriz \\(n \\times n\\) \u00e9 o fator de escala de volume da transforma\u00e7\u00e3o linear que a matriz representa: \\(|\\det A|\\) mostra quanto a transforma\u00e7\u00e3o multiplica comprimentos (2\u00d72), \u00e1reas (2\u00d72 e 3\u00d73 em contextos de 2D\/3D) ou volumes em geral. Um determinante igual a 3 significa que a transforma\u00e7\u00e3o triplica o volume; um determinante igual a 0 significa que a transforma\u00e7\u00e3o colapsa o espa\u00e7o em uma dimens\u00e3o inferior e perde informa\u00e7\u00e3o.<\/p>\n<p>Algebricamente, o determinante \u00e9 um teste r\u00e1pido de invertibilidade. Uma matriz quadrada tem inversa se, e somente se, \\(\\det A \\neq 0\\). Quando \\(\\det A = 0\\), a matriz \u00e9 chamada de <em>singular<\/em>: suas linhas (e colunas) s\u00e3o linearmente dependentes, os sistemas \\(Ax = b\\) n\u00e3o t\u00eam solu\u00e7\u00e3o \u00fanica, e \\(A^{-1}\\) n\u00e3o existe. \u00c9 por isso que o determinante aparece por toda parte, da resolu\u00e7\u00e3o de sistemas lineares (regra de Cramer) \u00e0 obten\u00e7\u00e3o de autovalores e ao c\u00e1lculo de produtos vetoriais.<\/p>\n<p>A forma de interpretar o n\u00famero depende do contexto. O valor exato importa para conferir contas \u00e0 m\u00e3o e para a teoria; o valor decimal \u00e9 pr\u00e1tico para estimativas. O <em>sinal<\/em> tamb\u00e9m carrega significado: um determinante negativo indica que a transforma\u00e7\u00e3o inverte a orienta\u00e7\u00e3o, como em um espelhamento. Note que os determinantes s\u00e3o definidos apenas para matrizes quadradas \u2014 uma matriz n\u00e3o quadrada simplesmente n\u00e3o tem determinante.<\/p>\n<h2>Como a calculadora faz o c\u00e1lculo: dois m\u00e9todos<\/h2>\n<p>Para matrizes 2\u00d72 e 3\u00d73, a calculadora usa a <strong>expans\u00e3o em cofatores<\/strong> ao longo da primeira linha e mostra todos os menores e produtos, incluindo uma verifica\u00e7\u00e3o cruzada pela regra de Sarrus para matrizes 3\u00d73. Para matrizes de 4\u00d74 a 6\u00d76, ela passa a usar a <strong>elimina\u00e7\u00e3o gaussiana<\/strong> com aritm\u00e9tica racional exata: somar a uma linha um m\u00faltiplo de outra deixa o determinante inalterado, enquanto cada troca de linhas inverte o seu sinal. O determinante \u00e9 ent\u00e3o o produto dos piv\u00f4s da matriz triangular superior resultante, multiplicado por \\((-1)\\) para cada troca realizada. Se n\u00e3o houver piv\u00f4 em alguma coluna, a matriz \u00e9 singular e o determinante \u00e9 exatamente 0.<\/p>\n<p>Como toda a aritm\u00e9tica \u00e9 feita com inteiros e fra\u00e7\u00f5es (e n\u00e3o com decimais de ponto flutuante), o resultado \u00e9 <em>exato<\/em> \u2014 n\u00e3o h\u00e1 erro de arredondamento, nem mesmo com fra\u00e7\u00f5es complicadas. Valores decimais com at\u00e9 12 casas s\u00e3o convertidos internamente em fra\u00e7\u00f5es exatas, de modo que 0,1 \u00e9 tratado como exatamente 1\/10, e n\u00e3o como um valor bin\u00e1rio aproximado.<\/p>\n<h2>Propriedades \u00fateis e limita\u00e7\u00f5es<\/h2>\n<ul>\n<li>\\(\\det(AB) = \\det A \\cdot \\det B\\), e \\(\\det(A^{-1}) = 1\/\\det A\\) para uma matriz invert\u00edvel.<\/li>\n<li>Trocar duas linhas (ou colunas) multiplica o determinante por \u22121; multiplicar uma linha por uma constante \\(k\\) multiplica o determinante por \\(k\\). Uma matriz com duas linhas iguais, ou com uma linha nula, tem determinante 0.<\/li>\n<li>\\(\\det A = \\det A^{T}\\), de modo que todo argumento v\u00e1lido para as linhas tamb\u00e9m vale para as colunas. Voc\u00ea pode explorar a pr\u00f3pria transposta com a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/calculadora-de-transposta-de-matriz\/\">calculadora de transposta de matriz<\/a>.<\/li>\n<li>A calculadora aceita apenas tamanhos de 2\u00d72 a 6\u00d76, com elementos de at\u00e9 12 casas decimais. Para matrizes maiores, use a elimina\u00e7\u00e3o combinada com a expans\u00e3o em cofatores, ou um software feito para grandes sistemas lineares.<\/li>\n<li>Um atalho que vale a pena memorizar: \\(\\det(kA) = k^{n} \\det A\\) para uma matriz \\(n \\times n\\), porque cada linha \u00e9 multiplicada por \\(k\\). A <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/multiplicacao-de-matriz-por-escalar\/\">calculadora de multiplica\u00e7\u00e3o de matriz por escalar<\/a> permite construir essas matrizes diretamente.<\/li>\n<li>Como um determinante diferente de zero garante solu\u00e7\u00e3o \u00fanica para um sistema linear, talvez voc\u00ea tamb\u00e9m queira resolver sistemas diretamente com a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/metodo-adicao\/\">calculadora de m\u00e9todo de elimina\u00e7\u00e3o<\/a>, ou comparar invariantes relacionados, como a soma dos elementos da diagonal, usando a <a href=\"https:\/\/wpcalc.com\/pt\/matematica\/traco-de-matriz\/\">calculadora de tra\u00e7o de matriz<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calcule o determinante exato de qualquer matriz quadrada de 2\u00d72 a 6\u00d76. Digite os elementos como inteiros, decimais ou fra\u00e7\u00f5es e obtenha o resultado exato, o valor decimal, a verifica\u00e7\u00e3o de singularidade e a solu\u00e7\u00e3o passo a passo.<\/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":[8],"tags":[54],"class_list":["post-3013","post","type-post","status-publish","format-standard","hentry","category-matematica","tag-matrizes"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/3013","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/comments?post=3013"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/posts\/3013\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/media?parent=3013"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/categories?post=3013"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/pt\/wp-json\/wp\/v2\/tags?post=3013"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}