{"id":236815,"date":"2022-05-29T17:12:00","date_gmt":"2022-05-29T15:12:00","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=236815"},"modified":"2025-09-29T11:39:43","modified_gmt":"2025-09-29T09:39:43","slug":"comment-calculer-la-comatrice","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/comment-calculer-la-comatrice\/","title":{"rendered":"Comment calculer la comatrice ?"},"content":{"rendered":"\n

En gal\u00e8re sur un exercice de calcul de la comatrice<\/strong> ? Rassure-toi, gr\u00e2ce \u00e0 ce cours enti\u00e8rement d\u00e9di\u00e9 \u00e0 la notion Comment calculer la comatrice<\/strong> ?<\/strong>, cette notion n’aura bient\u00f4t plus aucun secret pour toi ! Le 20\/20 est assur\u00e9 sur ta prochaine interrogation ! Et pour propulser tes comp\u00e9tences en alg\u00e8bre \u00e0 un niveau sup\u00e9rieur, apprends \u00e0 ma\u00eetriser la comatrice avec des cours en ligne de math\u00e9matiques<\/strong><\/a> interactifs.<\/p>\n\n\n\n

Comatrice<\/h2>\n\n\n\n

D\u00e9finition : Comatrice<\/h3>\n\n\n\n

<\/p>\nSoit \"A\in\mathcal{M}_n(\mathbb{K})\".\nLa matrice \"\big(\Delta_{i,j}\big)_{1\leq i,j\leq n}\" des cofacteurs de \"A\" est appel\u00e9e comatrice de \"A\" et est not\u00e9e \"\mathrm{Com}(A)\".\n\n\n\n

Proposition<\/h3>\n\n\n\n

<\/p>\nSoit \"A\in\mathcal{M}_n(\mathbb{K})\".
\nOn a \"{}^t\mathrm{Com}(A)\times A=A\times{}^t\mathrm{Com}(A)=\mathrm{det}(A).I_n\".\n\n\n\n

D\u00e9monstration<\/h3>\n\n\n\n

<\/p>\nOn note \"A=(a_{i,j})_{1\leq i,j\leq n}\".\nSoit \"(i,j)\in [[1,n]]^2\".
\nLe coefficient en position ligne \"i\" colonne \"j\" de \"{}^t \mathrm{Com}(A)\times A\" est \n\"\displaystyle\sum\limits_{k=1}^n \Delta_{k,i}a_{k,j}=\displaystyle\sum\limits_{k=1}^na_{k,j} \Delta_{k,i}.\"
\nOn note \"B_{i,j}\" la matrice obtenue en rempla\u00e7ant la \"i\"-\u00e8me colonne de \"A\" par la \"j\"-\u00e8me.
\n\nPar d\u00e9veloppement par rapport \u00e0 la \"i\"-\u00e8me colonne, on a \"\mathrm{det}(B_{i,j})=\displaystyle\sum\limits_{k=1}^na_{k,j}\Delta_{k,i}\".
\nSi \"i\neq j\", alors les colonnes \"i\" et \"j\" de \"B_{i,j}\" sont \u00e9gales, donc, \"\mathrm{det}(B_{i,j})=0\". Si \"i=j\", alors \"B_{i,j}=A\", donc \"\mathrm{det}(B_{i,j})=\mathrm{det}(A)\".
\nAinsi, \n

  <\/span>   <\/span>\"\[\sum\limits_{k=1}^n \Delta_{k,i}a_{k,j} = \left\lbrace\begin{array}{cl}<\/p>
\nDonc, \"{}^t\mathrm{Com}(A)\times A=\mathrm{det}(A).I_n\". L’\u00e9galit\u00e9 \"A\times {}^t\!\mathrm{Com}(A)=\mathrm{det}(A).I_n\" se montre de la m\u00eame mani\u00e8re.\n\n\n\n

On a alors imm\u00e9diatement le r\u00e9sultat suivant.<\/p>\n\n\n\n

Corollaire <\/h3>\n\n\n\n

<\/p>\nSoit \"A\in\mathrm{GL}_n(\mathbb{K})\". On a :\n

  <\/span>   <\/span>\"\[A^{-1} = \frac{1}{\mathrm{det}(A)}.{}^t\mathrm{Com}(A).\]\"<\/p>\n\n\n\n

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Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) \u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n\n <\/div>\n <\/section>\n<\/div>\n<\/div>\n\n\n

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\n 1.5\/5 - (2 votes) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

En gal\u00e8re sur un exercice de calcul de la comatrice ? Rassure-toi, gr\u00e2ce \u00e0 ce cours enti\u00e8rement d\u00e9di\u00e9 (…)<\/p>\n","protected":false},"author":158,"featured_media":244655,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-236815","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/236815","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=236815"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/236815\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/244655"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=236815"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=236815"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=236815"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}