{"id":6175,"date":"2026-11-02T11:57:01","date_gmt":"2026-11-02T08:57:01","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-determinant-matrice\/"},"modified":"2026-11-02T11:57:01","modified_gmt":"2026-11-02T08:57:01","slug":"calculateur-determinant-matrice","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-determinant-matrice\/","title":{"rendered":"Calculateur de d\u00e9terminant de matrice"},"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\">Calculer le d\u00e9terminant d\u2019une matrice carr\u00e9e<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"6175\" data-favorite-title=\"D\u00e9terminant\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-determinant-matrice\/\" data-favorite-excerpt=\"Calculez le d\u00e9terminant exact de toute matrice carr\u00e9e de 2\u00d72 \u00e0 6\u00d76. Saisissez des entiers, des nombres d\u00e9cimaux ou des fractions pour obtenir le r\u00e9sultat exact, sa valeur d\u00e9cimale, une v\u00e9rification de la singularit\u00e9 et la r\u00e9solution d\u00e9taill\u00e9e.\" aria-label=\"Ajouter aux favoris\" data-favorite-icon=\"icon icon-binary\">\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\/fr\/mathematiques\/calculateur-determinant-matrice\/\" method=\"POST\" class=\"calculator\" id=\"calculator-6175\" data-post=\"6175\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Taille de la matrice<\/label><div class=\"grid grid-cols-1\"><select id=\"field-1\" name=\"taille_de_la_matrice\"><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\">Remplir la grille avec<\/label><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"remplir_la_grille_avec\"><option value=\"custom\" selected>Les valeurs saisies ci-dessous<\/option><option value=\"random\">Matrice al\u00e9atoire \u00e0 coefficients entiers<\/option><option value=\"identity\">Matrice identit\u00e9<\/option><option value=\"zeros\">Effacer (tout mettre \u00e0 z\u00e9ro)<\/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\">Ligne 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\">Entiers, nombres d\u00e9cimaux (ex.\u00a0: 1,5) ou fractions (ex.\u00a0: 3\/4). Les cases vides valent 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\">Ligne 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\">Ligne 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\">Ligne 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\">Ligne 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\">Ligne 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=\"Recalculer\" id=\"calculate-button\" data-post=\"6175\">Calculer<\/button><button type=\"reset\">R\u00e9initialiser<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculer automatiquement<\/label><small>Si activ\u00e9, le r\u00e9sultat se met \u00e0 jour automatiquement d\u00e8s que vous modifiez une valeur.<\/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> R\u00e9sultats du calcul du d\u00e9terminant<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>D\u00e9terminant (valeur exacte)<\/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=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><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>D\u00e9terminant (valeur d\u00e9cimale)<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><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>La matrice est-elle singuli\u00e8re\u00a0?<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><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>\u00c9tapes du calcul<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><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\">Copier le lien vers ce calcul<\/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>Laisser un commentaire<\/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=\"6175\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Que pensez-vous de ce r\u00e9sultat\u00a0?<\/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\">J\u2019aime <\/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\">Utile <\/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\">Confus <\/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\">D\u00e9\u00e7u <\/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\">Inexact <\/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>Formule du d\u00e9terminant<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formule<\/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>Avec\u00a0:<\/strong><\/p>\n<ul>\n<li>\\(a_{ij}\\) \u2014 l\u2019\u00e9l\u00e9ment situ\u00e9 \u00e0 la ligne \\(i\\) et \u00e0 la colonne \\(j\\) de la matrice carr\u00e9e \\(A\\)<\/li>\n<li>\\(M_{1j}\\) \u2014 le <em>mineur<\/em>\u00a0: le d\u00e9terminant de la sous-matrice obtenue en supprimant la premi\u00e8re ligne et la colonne \\(j\\)<\/li>\n<li>\\((-1)^{1+j}M_{1j}\\) \u2014 le <em>cofacteur<\/em> de l\u2019\u00e9l\u00e9ment \\(a_{1j}\\)<\/li>\n<\/ul>\n<p>La premi\u00e8re ligne donne la r\u00e8gle des diagonales pour une matrice 2\u00d72. La deuxi\u00e8me d\u00e9veloppe le d\u00e9terminant d\u2019une matrice 3\u00d73 suivant sa premi\u00e8re ligne, \u00e0 l\u2019aide des cofacteurs (d\u00e9veloppement de Laplace). La derni\u00e8re donne la formule g\u00e9n\u00e9rale, valable pour toute matrice \\(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>D\u00e9terminant : exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple\u00a0:<\/strong> calculons le d\u00e9terminant de la matrice 3\u00d73 suivante\u00a0:<\/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>R\u00e9sultat\u00a0:<\/strong> \\(\\det A = 4\\). Le d\u00e9terminant n\u2019est pas nul\u00a0: la matrice est donc inversible. Vous pouvez retrouver ce r\u00e9sultat en lan\u00e7ant le calcul avec les valeurs propos\u00e9es par d\u00e9faut.<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur de d\u00e9terminant<\/h2>\n<ol>\n<li>Dans le menu <strong>Taille de la matrice<\/strong>, choisissez 2 \u00d7 2, 3 \u00d7 3, 4 \u00d7 4, 5 \u00d7 5 ou 6 \u00d7 6. Seuls les champs correspondant \u00e0 la taille choisie sont pris en compte.<\/li>\n<li>Dans le menu <strong>Remplir la grille avec<\/strong>, choisissez comment renseigner la matrice\u00a0: <em>Utiliser les valeurs saisies ci-dessous<\/em>, <em>Matrice al\u00e9atoire d\u2019entiers<\/em> (une nouvelle matrice est g\u00e9n\u00e9r\u00e9e \u00e0 chaque s\u00e9lection), <em>Matrice identit\u00e9<\/em> ou <em>Effacer (tout mettre \u00e0 z\u00e9ro)<\/em>.<\/li>\n<li>Saisissez les \u00e9l\u00e9ments dans les champs a<sub>11<\/sub>, a<sub>12<\/sub>, \u2026 Les entiers, les nombres d\u00e9cimaux comme 1,5 et les fractions comme 3\/4 sont accept\u00e9s. Une case vide vaut 0.<\/li>\n<li>Consultez les r\u00e9sultats\u00a0: <strong>D\u00e9terminant (valeur exacte)<\/strong> donne le r\u00e9sultat sous forme d\u2019entier ou de fraction exacte, <strong>D\u00e9terminant (valeur d\u00e9cimale)<\/strong> en donne la forme d\u00e9cimale, <strong>La matrice est-elle singuli\u00e8re\u00a0?<\/strong> indique si elle est inversible et <strong>\u00c9tapes de la r\u00e9solution<\/strong> d\u00e9taille le calcul.<\/li>\n<\/ol>\n<h2>Qu\u2019est-ce que le d\u00e9terminant d\u2019une matrice\u00a0?<\/h2>\n<p>Le d\u00e9terminant est un nombre qui r\u00e9sume une propri\u00e9t\u00e9 g\u00e9om\u00e9trique et alg\u00e9brique importante d\u2019une matrice carr\u00e9e. Sur le plan g\u00e9om\u00e9trique, la valeur absolue du d\u00e9terminant d\u2019une matrice \\(n \\times n\\) est le facteur par lequel la transformation lin\u00e9aire repr\u00e9sent\u00e9e par cette matrice modifie les volumes\u00a0: \\(|\\det A|\\) indique dans quelle mesure elle multiplie les longueurs (2\u00d72), les aires (2\u00d72 et 3\u00d73 dans des contextes en 2D ou 3D) ou, plus g\u00e9n\u00e9ralement, les volumes. Un d\u00e9terminant de 3 signifie que la transformation triple le volume\u00a0; un d\u00e9terminant de 0 signifie qu\u2019elle ram\u00e8ne l\u2019espace \u00e0 une dimension inf\u00e9rieure, avec une perte d\u2019information.<\/p>\n<p>Sur le plan alg\u00e9brique, le d\u00e9terminant permet de savoir rapidement si une matrice est inversible. Une matrice carr\u00e9e poss\u00e8de une inverse si et seulement si \\(\\det A \\neq 0\\). Lorsque \\(\\det A = 0\\), la matrice est dite <em>singuli\u00e8re<\/em>\u00a0: ses lignes (et ses colonnes) sont lin\u00e9airement d\u00e9pendantes, les syst\u00e8mes \\(Ax = b\\) n\u2019ont pas de solution unique et \\(A^{-1}\\) n\u2019existe pas. C\u2019est pourquoi le d\u00e9terminant intervient aussi bien dans la r\u00e9solution des syst\u00e8mes lin\u00e9aires (r\u00e8gle de Cramer) que dans la recherche des valeurs propres et le calcul des produits vectoriels.<\/p>\n<p>L\u2019interpr\u00e9tation du r\u00e9sultat d\u00e9pend du contexte. Sa valeur exacte est utile pour v\u00e9rifier un calcul \u00e0 la main et pour la th\u00e9orie\u00a0; sa valeur d\u00e9cimale facilite les estimations. Son <em>signe<\/em> a \u00e9galement un sens\u00a0: un d\u00e9terminant n\u00e9gatif indique que la transformation inverse l\u2019orientation, comme dans une r\u00e9flexion par un miroir. Enfin, le d\u00e9terminant n\u2019est d\u00e9fini que pour les matrices carr\u00e9es\u00a0: une matrice non carr\u00e9e n\u2019en a pas.<\/p>\n<h2>Comment fonctionne le calculateur\u00a0: deux m\u00e9thodes<\/h2>\n<p>Pour les matrices 2\u00d72 et 3\u00d73, le calculateur utilise le <strong>d\u00e9veloppement suivant la premi\u00e8re ligne \u00e0 l\u2019aide des cofacteurs<\/strong>. Il affiche chaque mineur et chaque produit et, pour les matrices 3\u00d73, v\u00e9rifie aussi le r\u00e9sultat avec la r\u00e8gle de Sarrus. Pour les matrices de 4\u00d74 \u00e0 6\u00d76, il passe \u00e0 <strong>l\u2019\u00e9limination de Gauss<\/strong> en calculant avec des fractions exactes\u00a0: ajouter \u00e0 une ligne un multiple d\u2019une autre ne change pas le d\u00e9terminant, tandis que permuter deux lignes en inverse le signe. Le d\u00e9terminant est alors le produit des pivots de la matrice triangulaire sup\u00e9rieure obtenue, multipli\u00e9 par \\((-1)\\) pour chaque permutation effectu\u00e9e. Si aucun pivot ne peut \u00eatre trouv\u00e9 dans une colonne, la matrice est singuli\u00e8re et son d\u00e9terminant vaut exactement 0.<\/p>\n<p>Tous les calculs sont effectu\u00e9s avec des entiers et des fractions, et non avec des nombres d\u00e9cimaux en virgule flottante. Le r\u00e9sultat est donc <em>exact<\/em>, sans erreur d\u2019arrondi, m\u00eame avec des fractions difficiles \u00e0 manipuler. Les nombres d\u00e9cimaux comportant jusqu\u2019\u00e0 12 chiffres apr\u00e8s la virgule sont convertis en fractions exactes\u00a0: 0,1 est ainsi trait\u00e9 comme exactement 1\/10, et non comme une valeur binaire approch\u00e9e.<\/p>\n<h2>Propri\u00e9t\u00e9s utiles et limites<\/h2>\n<ul>\n<li>\\(\\det(AB) = \\det A \\cdot \\det B\\), et \\(\\det(A^{-1}) = 1\/\\det A\\) lorsque la matrice est inversible.<\/li>\n<li>Permuter deux lignes (ou deux colonnes) multiplie le d\u00e9terminant par \u22121\u00a0; multiplier une ligne par une constante \\(k\\) multiplie le d\u00e9terminant par \\(k\\). Une matrice ayant deux lignes identiques ou une ligne nulle a un d\u00e9terminant nul.<\/li>\n<li>\\(\\det A = \\det A^{T}\\)\u00a0: les propri\u00e9t\u00e9s relatives aux lignes s\u2019appliquent donc aussi aux colonnes. Vous pouvez \u00e9tudier la transpos\u00e9e avec le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-transposee-matrice\/\">calculateur de transpos\u00e9e d\u2019une matrice<\/a>.<\/li>\n<li>Le calculateur accepte uniquement les matrices de 2\u00d72 \u00e0 6\u00d76, avec des \u00e9l\u00e9ments comportant jusqu\u2019\u00e0 12 chiffres apr\u00e8s la virgule. Pour les matrices plus grandes, utilisez l\u2019\u00e9limination et le d\u00e9veloppement par les cofacteurs, ou un logiciel adapt\u00e9 aux grands syst\u00e8mes lin\u00e9aires.<\/li>\n<li>Une propri\u00e9t\u00e9 pratique \u00e0 retenir\u00a0: \\(\\det(kA) = k^{n} \\det A\\) pour une matrice \\(n \\times n\\), car chaque ligne est multipli\u00e9e par \\(k\\). Le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/multiplication-matrice-par-scalaire\/\">calculateur de multiplication d\u2019une matrice par un scalaire<\/a> permet de construire directement de telles matrices.<\/li>\n<li>Un d\u00e9terminant non nul garantissant qu\u2019un syst\u00e8me lin\u00e9aire poss\u00e8de une solution unique, vous pouvez aussi r\u00e9soudre directement des syst\u00e8mes avec le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/systemes-par-elimination\/\">calculateur de r\u00e9solution par \u00e9limination<\/a>, ou comparer des invariants li\u00e9s, comme la somme des \u00e9l\u00e9ments diagonaux, avec le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calcul-trace-matrice\/\">calculateur de trace d\u2019une matrice<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculez le d\u00e9terminant exact de toute matrice carr\u00e9e de 2\u00d72 \u00e0 6\u00d76. Saisissez des entiers, des nombres d\u00e9cimaux ou des fractions pour obtenir le r\u00e9sultat exact, sa valeur d\u00e9cimale, une v\u00e9rification de la singularit\u00e9 et la r\u00e9solution d\u00e9taill\u00e9e.<\/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":[2],"tags":[24],"class_list":["post-6175","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-matrices"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/6175","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/comments?post=6175"}],"version-history":[{"count":0,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/6175\/revisions"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=6175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=6175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=6175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}