{"id":201,"date":"2026-10-03T06:04:16","date_gmt":"2026-10-03T03:04:16","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/?p=201"},"modified":"2026-10-03T06:04:16","modified_gmt":"2026-10-03T03:04:16","slug":"produit-determinants","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/mathematiques\/produit-determinants\/","title":{"rendered":"Calculateur de produit de d\u00e9terminants"},"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\">Calculez le produit de deux d\u00e9terminants<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"201\" data-favorite-title=\"Produit de d\u00e9terminants\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/mathematiques\/produit-determinants\/\" data-favorite-excerpt=\"Saisissez deux matrices carr\u00e9es (2 \u00d7 2 ou 3 \u00d7 3) et obtenez instantan\u00e9ment det(A), det(B) et leur produit. Ce calcul repose sur la propri\u00e9t\u00e9 multiplicative des d\u00e9terminants\u00a0: det(AB) = det(A)\u00b7det(B).\" 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\/produit-determinants\/\" method=\"POST\" class=\"calculator\" id=\"calculator-201\" data-post=\"201\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field is-single\" 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\">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\">Matrice 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\">Matrice 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=\"Recalculer\" id=\"calculate-button\" data-post=\"201\">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> Vos r\u00e9sultats<\/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=\"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\" 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=\"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\" id=\"output-3\"><span class=\"field-title\"><span>|A| \u00d7 |B|<\/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><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=\"201\" 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 produit de d\u00e9terminants<\/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\\[ \\det(A \\cdot B) = \\det(A) \\times \\det(B) \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>O\u00f9&nbsp;:<\/strong><\/p>\n<ul>\n<li>\\(A\\) et \\(B\\) \u2014 des matrices carr\u00e9es de m\u00eame taille (2 \u00d7 2 ou 3 \u00d7 3 ici)<\/li>\n<li>\\(\\det(A)\\) \u2014 le d\u00e9terminant de la matrice \\(A\\)<\/li>\n<li>\\(\\det(B)\\) \u2014 le d\u00e9terminant de la matrice \\(B\\)<\/li>\n<li>\\(A \\cdot B\\) \u2014 le produit matriciel de \\(A\\) et \\(B\\)<\/li>\n<\/ul>\n<p>Le d\u00e9terminant d\u2019un produit est \u00e9gal au produit des d\u00e9terminants. Ainsi, au lieu de multiplier deux matrices puis de calculer le d\u00e9terminant de la matrice obtenue, vous pouvez calculer chaque d\u00e9terminant s\u00e9par\u00e9ment et multiplier les deux nombres&nbsp;: le r\u00e9sultat est identique.<\/p>\n<p>Pour chaque d\u00e9terminant, le calculateur utilise les formules classiques&nbsp;:<\/p>\n<p>\\[ \\det \\begin{bmatrix} a &amp; b \\\\ c &amp; d \\end{bmatrix} = ad - bc \\]<br \/>\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}) \\]<br \/>\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>Produit de d\u00e9terminants \u2013 exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple&nbsp;:<\/strong> calculons le produit des d\u00e9terminants de deux matrices 3 \u00d7 3.<\/p>\n<p>\\[ A = \\begin{bmatrix} 2 &amp; 1 &amp; 3 \\\\ 0 &amp; 4 &amp; 5 \\\\ 1 &amp; 2 &amp; 6 \\end{bmatrix}, \\quad B = \\begin{bmatrix} 1 &amp; 2 &amp; 0 \\\\ 3 &amp; 1 &amp; 4 \\\\ 0 &amp; 5 &amp; 2 \\end{bmatrix} \\]<\/p>\n<p>\\[ \\det(A) = 2(4 \\cdot 6 - 5 \\cdot 2) - 1(0 \\cdot 6 - 5 \\cdot 1) + 3(0 \\cdot 2 - 4 \\cdot 1) = 2(14) - 1(-5) + 3(-4) = 28 + 5 - 12 = 21 \\]<\/p>\n<p>\\[ \\det(B) = 1(1 \\cdot 2 - 4 \\cdot 5) - 2(3 \\cdot 2 - 4 \\cdot 0) + 0(3 \\cdot 5 - 1 \\cdot 0) = 1(-18) - 2(6) + 0 = -30 \\]<\/p>\n<p>\\[ \\det(A) \\times \\det(B) = 21 \\times (-30) = -630 \\]<\/p>\n<p><strong>R\u00e9sultat&nbsp;:<\/strong> \\(|A| = 21\\), \\(|B| = -30\\) et \\(|A| \\times |B| = -630\\).<\/p>\n<p>C\u2019est aussi le d\u00e9terminant du produit matriciel \\(A \\cdot B\\) : multiplier d\u2019abord les deux matrices puis calculer le d\u00e9terminant donnerait la m\u00eame valeur, \\(-630\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur de produit de d\u00e9terminants<\/h2>\n<ol>\n<li>S\u00e9lectionnez la <strong>taille des matrices<\/strong> (2 \u00d7 2 ou 3 \u00d7 3) dans le menu d\u00e9roulant en haut de la page. Les champs de saisie s\u2019adaptent automatiquement.<\/li>\n<li>Saisissez les \u00e9l\u00e9ments de la <strong>matrice A<\/strong> dans les champs libell\u00e9s de a\u2081\u2081 \u00e0 a\u2083\u2083 (pour une matrice 2 \u00d7 2, seuls a\u2081\u2081, a\u2081\u2082, a\u2082\u2081 et a\u2082\u2082 sont affich\u00e9s).<\/li>\n<li>Saisissez les \u00e9l\u00e9ments de la <strong>matrice B<\/strong> dans les champs libell\u00e9s de b\u2081\u2081 \u00e0 b\u2083\u2083.<\/li>\n<li>Lisez les r\u00e9sultats&nbsp;: <strong>|A|<\/strong> est le d\u00e9terminant de la matrice A, <strong>|B|<\/strong> celui de la matrice B et <strong>|A| \u00d7 |B|<\/strong> leur produit.<\/li>\n<\/ol>\n<h2>Qu\u2019est-ce que le produit de d\u00e9terminants ?<\/h2>\n<p>Le d\u00e9terminant est un nombre associ\u00e9 \u00e0 une matrice carr\u00e9e. G\u00e9om\u00e9triquement, pour une matrice 2 \u00d7 2, il repr\u00e9sente le facteur de changement d\u2019aire de la transformation d\u00e9crite par la matrice ; pour une matrice 3 \u00d7 3, il repr\u00e9sente un facteur de changement de volume. En alg\u00e8bre lin\u00e9aire, il intervient partout&nbsp;: dans la r\u00e9solution de syst\u00e8mes d\u2019\u00e9quations, la recherche de matrices inverses, le calcul des valeurs propres et la v\u00e9rification de l\u2019inversibilit\u00e9 d\u2019une matrice.<\/p>\n<p>Le produit de d\u00e9terminants repose sur leur propri\u00e9t\u00e9 multiplicative&nbsp;: si vous multipliez deux matrices carr\u00e9es de m\u00eame taille puis calculez le d\u00e9terminant du r\u00e9sultat, vous obtenez exactement le m\u00eame nombre qu\u2019en calculant s\u00e9par\u00e9ment chaque d\u00e9terminant puis en multipliant les deux nombres. En notation math\u00e9matique&nbsp;:<\/p>\n<p>\\[ \\det(A \\cdot B) = \\det(A) \\cdot \\det(B) \\]<\/p>\n<p>C\u2019est une propri\u00e9t\u00e9 remarquable&nbsp;: la multiplication matricielle est une op\u00e9ration complexe qui fait intervenir des sommes de produits de nombreux \u00e9l\u00e9ments, tandis que le d\u00e9terminant du produit se r\u00e9duit \u00e0 une simple multiplication de deux nombres. Elle s\u2019applique \u00e0 un nombre quelconque de facteurs : \\(\\det(ABC) = \\det(A)\\det(B)\\det(C)\\) et, plus g\u00e9n\u00e9ralement, \\(\\det(A^n) = (\\det A)^n\\). C\u2019est ce que donne le calculateur <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/matrice-au-carre\/\">Carr\u00e9 d\u2019une matrice<\/a> pour le d\u00e9terminant d\u2019une matrice \u00e9lev\u00e9e au carr\u00e9.<\/p>\n<p>L\u2019interpr\u00e9tation du r\u00e9sultat est simple&nbsp;: le produit \\(\\det(A) \\cdot \\det(B)\\) donne le facteur de changement d\u2019\u00e9chelle combin\u00e9 lorsqu\u2019on applique la transformation \\(A\\), puis la transformation \\(B\\). Un produit positif conserve l\u2019orientation, un produit n\u00e9gatif l\u2019inverse et un produit nul signifie qu\u2019au moins une des matrices est singuli\u00e8re (non inversible) et r\u00e9duit l\u2019espace \u00e0 une dimension inf\u00e9rieure. Attention, les d\u00e9terminants ne s\u2019additionnent pas de la m\u00eame mani\u00e8re : en g\u00e9n\u00e9ral, \\(\\det(A) + \\det(B) \\neq \\det(A + B)\\). C\u2019est pourquoi le calculateur de <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/somme-des-determinants\/\">somme de d\u00e9terminants<\/a> et celui-ci sont deux outils distincts.<\/p>\n<h2>Conseils pratiques et limites<\/h2>\n<ul>\n<li><strong>M\u00eame taille requise&nbsp;:<\/strong> la propri\u00e9t\u00e9 \\(\\det(A \\cdot B) = \\det(A)\\det(B)\\) n\u2019a de sens que si \\(A\\) et \\(B\\) sont toutes deux des matrices carr\u00e9es de m\u00eame taille. Ce calculateur prend en charge les matrices 2 \u00d7 2 et 3 \u00d7 3.<\/li>\n<li><strong>Un d\u00e9terminant nul est important&nbsp;:<\/strong> si l\u2019un des d\u00e9terminants vaut 0, le produit vaut 0 et la matrice correspondante n\u2019a pas d\u2019inverse. Vous pouvez v\u00e9rifier directement l\u2019inversibilit\u00e9 avec le calculateur de <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-inverse-matrice-3x3\/\">matrice inverse 3\u00d73<\/a>.<\/li>\n<li><strong>L\u2019ordre n\u2019a pas d\u2019importance pour le produit des d\u00e9terminants&nbsp;:<\/strong> puisque la multiplication ordinaire est commutative, \\(\\det(A)\\det(B) = \\det(B)\\det(A)\\), m\u00eame si les produits matriciels \\(AB\\) et \\(BA\\) sont g\u00e9n\u00e9ralement diff\u00e9rents.<\/li>\n<li><strong>Ce n\u2019est pas le d\u00e9terminant d\u2019une somme&nbsp;:<\/strong> saisir deux matrices ici ne vous donnera pas \\(\\det(A + B)\\) ; il n\u2019existe pas de formule simple pour cette valeur.<\/li>\n<li><strong>Les nombres n\u00e9gatifs et d\u00e9cimaux sont accept\u00e9s&nbsp;:<\/strong> le calculateur accepte tous les nombres r\u00e9els, y compris les valeurs n\u00e9gatives et les nombres d\u00e9cimaux, et arrondit les r\u00e9sultats affich\u00e9s.<\/li>\n<li><strong>Dimensions sup\u00e9rieures&nbsp;:<\/strong> la m\u00eame propri\u00e9t\u00e9 s\u2019applique aux matrices 4 \u00d7 4 et plus grandes, mais leurs d\u00e9terminants doivent alors \u00eatre calcul\u00e9s par d\u00e9veloppement par cofacteurs ou par r\u00e9duction par lignes, ce que cet outil ne prend pas en charge.<\/li>\n<\/ul>\n<h2>R\u00e9capitulatif : principales propri\u00e9t\u00e9s des d\u00e9terminants<\/h2>\n<table>\n<thead>\n<tr>\n<th>Propri\u00e9t\u00e9<\/th>\n<th>Formule<\/th>\n<th>Condition<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Multiplicativit\u00e9<\/td>\n<td>\\(\\det(AB) = \\det(A)\\det(B)\\)<\/td>\n<td>\\(A, B\\) carr\u00e9es, de m\u00eame taille<\/td>\n<\/tr>\n<tr>\n<td>Transpos\u00e9e<\/td>\n<td>\\(\\det(A^T) = \\det(A)\\)<\/td>\n<td>toute matrice carr\u00e9e<\/td>\n<\/tr>\n<tr>\n<td>Multiplication par un scalaire<\/td>\n<td>\\(\\det(kA) = k^n \\det(A)\\)<\/td>\n<td>\\(A\\) est de taille \\(n \\times n\\)<\/td>\n<\/tr>\n<tr>\n<td>Matrice identit\u00e9<\/td>\n<td>\\(\\det(I) = 1\\)<\/td>\n<td>toute taille<\/td>\n<\/tr>\n<tr>\n<td>Inverse<\/td>\n<td>\\(\\det(A^{-1}) = 1\/\\det(A)\\)<\/td>\n<td>\\(\\det(A) \\neq 0\\)<\/td>\n<\/tr>\n<tr>\n<td>Matrice singuli\u00e8re<\/td>\n<td>\\(\\det(A) = 0 \\iff A\\) non inversible<\/td>\n<td>toute matrice carr\u00e9e<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Si vous devez calculer le produit matriciel complet \\(A \\cdot B\\) \u00e9l\u00e9ment par \u00e9l\u00e9ment avant de vous int\u00e9resser aux d\u00e9terminants, utilisez le calculateur de <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/multiplication-matrices\/\">multiplication de matrices<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Saisissez deux matrices carr\u00e9es (2 \u00d7 2 ou 3 \u00d7 3) et obtenez instantan\u00e9ment det(A), det(B) et leur produit. Ce calcul repose sur la propri\u00e9t\u00e9 multiplicative des d\u00e9terminants\u00a0: det(AB) = det(A)\u00b7det(B).<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[24],"class_list":["post-201","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\/201","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=201"}],"version-history":[{"count":6,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/201\/revisions"}],"predecessor-version":[{"id":5166,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/201\/revisions\/5166"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=201"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=201"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=201"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}