{"id":1552,"date":"2026-09-08T08:06:58","date_gmt":"2026-09-08T08:06:58","guid":{"rendered":"https:\/\/maroczain.com\/scolaire.maroczain.com\/?page_id=1552"},"modified":"2026-09-08T08:06:58","modified_gmt":"2026-09-08T08:06:58","slug":"__trashed-2","status":"publish","type":"page","link":"https:\/\/maroczain.com\/scolaire.maroczain.com\/__trashed-2\/","title":{"rendered":"Mesurer. D\u00e9cider. Inverser."},"content":{"rendered":"\n&#8220;`html id=&#8221;mza-determinants-13&#8243;\n<div id=\"mza-determinants-cpge\">\n\n<style>\n#mza-determinants-cpge,#mza-determinants-cpge *{box-sizing:border-box}\n#mza-determinants-cpge{\n --navy:#06172c;--navy2:#0d416d;--gold:#c99a36;--gold2:#efd68c;\n --paper:#f5f6f8;--white:#fff;--text:#18283a;--muted:#687789;--line:#dfe5eb;\n font-family:Arial,Helvetica,sans-serif;background:var(--paper);color:var(--text);overflow:hidden\n}\n#mza-determinants-cpge a{text-decoration:none;color:inherit}\n#mza-determinants-cpge .wrap{max-width:1200px;margin:auto;padding:0 24px}\n#mza-determinants-cpge section{padding:68px 0}\n#mza-determinants-cpge .hero{\n color:#fff;padding:72px 0;\n background:\n radial-gradient(circle at 84% 15%,rgba(213,169,68,.28),transparent 28%),\n linear-gradient(135deg,#041224,#082b4c 65%,#10517e)\n}\n#mza-determinants-cpge .badge{\n display:inline-block;padding:8px 13px;border-radius:999px;\n border:1px solid 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.grid,#mza-determinants-cpge .method{grid-template-columns:1fr}\n}\n<\/style>\n\n<header class=\"hero\">\n<div class=\"wrap\">\n<span class=\"badge\">COURS 13 \u2022 D\u00c9TERMINANTS<\/span>\n\n<h1>Mesurer.<br>D\u00e9cider. Inverser.<\/h1>\n\n<p class=\"lead\">\nLe d\u00e9terminant permet de d\u00e9tecter imm\u00e9diatement l&#8217;inversibilit\u00e9 d&#8217;une matrice,\nmesurer un facteur de volume, tester l&#8217;ind\u00e9pendance de vecteurs\net simplifier de nombreuses questions d&#8217;alg\u00e8bre lin\u00e9aire.\n<\/p>\n\n<div class=\"actions\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda COMMENCER<\/a>\n<a href=\"#methode\" class=\"btn glass\">\ud83e\udde0 M\u00c9THODE<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 QCM<\/a>\n<\/div>\n<\/div>\n<\/header>\n\n<section id=\"cours\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">1. D\u00c9TERMINANT EN DIMENSION 2<\/span>\n<h2>Le premier calcul fondamental.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Matrice 2\u00d72<\/h3>\n\n<div class=\"formula\">\nA = [[a,b],[c,d]]\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>D\u00e9terminant<\/h3>\n\n<div class=\"formula\">\ndet(A)=ad-bc\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Inversibilit\u00e9<\/h3>\n\n<div class=\"formula\">\nA inversible \u21d4 det(A) \u2260 0\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Interpr\u00e9tation<\/h3>\n\n<p>\nSi det(A)=0, les colonnes \u2014 ou les lignes \u2014 sont lin\u00e9airement d\u00e9pendantes.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">2. PROPRI\u00c9T\u00c9S FONDAMENTALES<\/span>\n<h2>Calculer sans d\u00e9velopper inutilement.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Produit<\/h3>\n\n<div class=\"formula\">\ndet(AB)=det(A)det(B)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Identit\u00e9<\/h3>\n\n<div class=\"formula\">\ndet(I)=1\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Inverse<\/h3>\n\n<p>Si A est inversible :<\/p>\n\n<div class=\"formula\">\ndet(A\u207b\u00b9)=1\/det(A)\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Transpos\u00e9e<\/h3>\n\n<div class=\"formula\">\ndet(A\u1d40)=det(A)\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">3. OP\u00c9RATIONS SUR LES LIGNES<\/span>\n<h2>Comprendre l&#8217;effet sur le d\u00e9terminant.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>\u00c9change de deux lignes<\/h3>\n\n<p>\nLe d\u00e9terminant change de signe.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Multiplication d&#8217;une ligne<\/h3>\n\n<p>\nSi une ligne est multipli\u00e9e par \u03bb :\n<\/p>\n\n<div class=\"formula\">\ndet \u2192 \u03bb det\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Ajout d&#8217;un multiple d&#8217;une ligne<\/h3>\n\n<p>\nAjouter \u00e0 une ligne un multiple d&#8217;une autre ne change pas le d\u00e9terminant.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Deux lignes \u00e9gales<\/h3>\n\n<div class=\"formula\">\ndet(A)=0\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"methode\" class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA DETERMINANT ENGINE<\/span>\n<h2>Six r\u00e9flexes avant de d\u00e9velopper.<\/h2>\n<\/div>\n\n<div class=\"method\">\n<div><b>01<\/b><span>STRUCTURE<\/span><\/div>\n<div><b>02<\/b><span>Z\u00c9ROS<\/span><\/div>\n<div><b>03<\/b><span>OP\u00c9RATIONS LIGNES<\/span><\/div>\n<div><b>04<\/b><span>FACTORISER<\/span><\/div>\n<div><b>05<\/b><span>D\u00c9VELOPPER<\/span><\/div>\n<div><b>06<\/b><span>INTERPR\u00c9TER<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">4. D\u00c9VELOPPEMENT PAR COFACTEURS<\/span>\n<h2>Exploiter une ligne ou une colonne simple.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Mineur<\/h3>\n\n<p>\nOn supprime une ligne et une colonne pour obtenir une matrice de taille inf\u00e9rieure.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Cofacteur<\/h3>\n\n<div class=\"formula\">\nC<sub>ij<\/sub>=(-1)<sup>i+j<\/sup>M<sub>ij<\/sub>\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>D\u00e9veloppement selon la ligne i<\/h3>\n\n<div class=\"formula\">\ndet(A)=\u03a3 a<sub>ij<\/sub>C<sub>ij<\/sub>\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>Strat\u00e9gie<\/h3>\n\n<div class=\"tip\">\nD\u00e9veloppez de pr\u00e9f\u00e9rence selon une ligne ou une colonne contenant beaucoup de z\u00e9ros.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">5. MATRICES TRIANGULAIRES<\/span>\n<h2>Le cas le plus rapide.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nPour une matrice triangulaire :\n<\/p>\n\n<div class=\"formula\">\ndet(A)=a\u2081\u2081a\u2082\u2082&#8230;a\u2099\u2099\n<\/div>\n\n<p>\nLe d\u00e9terminant est simplement le produit des coefficients diagonaux.\n<\/p>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">6. LIEN AVEC LES BASES<\/span>\n<h2>Tester l&#8217;ind\u00e9pendance de vecteurs.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Famille de n vecteurs dans K\u207f<\/h3>\n\n<p>\nOn place leurs coordonn\u00e9es en colonnes d&#8217;une matrice A.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Crit\u00e8re<\/h3>\n\n<div class=\"formula\">\ndet(A) \u2260 0\n<\/div>\n\n<p>\n\u00e9quivaut \u00e0 dire que la famille est une base de K\u207f.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Si det(A)=0<\/h3>\n\n<p>\nLes vecteurs sont lin\u00e9airement d\u00e9pendants.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>R\u00e9flexe<\/h3>\n\n<div class=\"tip\">\nEn dimension n, le d\u00e9terminant donne un test extr\u00eamement rapide pour une famille de n vecteurs.\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">7. D\u00c9TERMINANT D&#8217;UN ENDOMORPHISME<\/span>\n<h2>Une quantit\u00e9 ind\u00e9pendante du choix de base.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<p>\nSi f est un endomorphisme de E de dimension finie, son d\u00e9terminant est le d\u00e9terminant\nde n&#8217;importe quelle matrice repr\u00e9sentant f dans une base.\n<\/p>\n\n<div class=\"formula\">\ndet(f)=det(Mat<sub>B<\/sub>(f))\n<\/div>\n\n<p>\nCette valeur ne d\u00e9pend pas de la base choisie.\n<\/p>\n\n<\/div>\n\n<div class=\"grid\" style=\"margin-top:16px\">\n\n<div class=\"card\">\n<h3>Injectivit\u00e9<\/h3>\n\n<div class=\"formula\">\ndet(f) \u2260 0 \u21d2 f injective\n<\/div>\n<\/div>\n\n<div class=\"card\">\n<h3>En dimension finie<\/h3>\n\n<div class=\"formula\">\ndet(f) \u2260 0 \u21d4 f bijective\n<\/div>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">8. INTERPR\u00c9TATION G\u00c9OM\u00c9TRIQUE<\/span>\n<h2>Aire, volume et orientation.<\/h2>\n<\/div>\n\n<div class=\"grid\">\n\n<div class=\"card\">\n<h3>Dimension 2<\/h3>\n\n<p>\nLa valeur absolue du d\u00e9terminant mesure le facteur d&#8217;aire induit par la transformation.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Dimension 3<\/h3>\n\n<p>\nLa valeur absolue du d\u00e9terminant mesure le facteur de volume.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>Signe<\/h3>\n\n<p>\nLe signe du d\u00e9terminant renseigne sur l&#8217;orientation.\n<\/p>\n<\/div>\n\n<div class=\"card\">\n<h3>D\u00e9terminant nul<\/h3>\n\n<p>\nUne dimension est \u00ab \u00e9cras\u00e9e \u00bb : l&#8217;image est contenue dans un sous-espace de dimension plus petite.\n<\/p>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section id=\"exercices\" style=\"background:#efede7\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">9. EXERCICES PROGRESSIFS<\/span>\n<h2>Choisir le calcul le plus efficace.<\/h2>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 1<\/span>\n\n<h3>Exercice 1 \u2014 Dimension 2<\/h3>\n\n<p>Calculer :<\/p>\n\n<div class=\"formula\">\ndet([[2,3],[1,4]])\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDetHint('detHint1')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"detHint1\" class=\"hint\">\nUtilisez directement la formule ad-bc.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 2<\/span>\n\n<h3>Exercice 2 \u2014 Triangulaire<\/h3>\n\n<p>Calculer :<\/p>\n\n<div class=\"formula\">\ndet([[2,1,4],[0,3,5],[0,0,-2]])\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDetHint('detHint2')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"detHint2\" class=\"hint\">\nNe d\u00e9veloppez pas : observez la forme de la matrice.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU 3<\/span>\n\n<h3>Exercice 3 \u2014 Param\u00e8tre<\/h3>\n\n<p>\nD\u00e9terminer les valeurs de \u03bb pour lesquelles la matrice\n<\/p>\n\n<div class=\"formula\">\nA=[[1,\u03bb],[2,4]]\n<\/div>\n\n<p>est inversible.<\/p>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDetHint('detHint3')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"detHint3\" class=\"hint\">\nCalculez le d\u00e9terminant en fonction de \u03bb puis imposez qu&#8217;il soit non nul.\n<\/div>\n<\/div>\n\n<div class=\"exercise\">\n<span class=\"level\">NIVEAU PR\u00c9PA<\/span>\n\n<h3>Exercice 4 \u2014 Famille de vecteurs<\/h3>\n\n<p>\n\u00c9tudier si les vecteurs suivants forment une base de \u211d\u00b3 :\n<\/p>\n\n<div class=\"formula\">\nu=(1,1,0)<br>\nv=(0,1,1)<br>\nw=(1,0,1)\n<\/div>\n\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDetHint('detHint4')\">\ud83d\udca1 INDICE<\/button>\n\n<div id=\"detHint4\" class=\"hint\">\nConstruisez la matrice ayant u, v et w pour colonnes.\n<\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">10. PROBLEM LAB<\/span>\n<h2>Mission : d\u00e9tecter une valeur critique.<\/h2>\n<\/div>\n\n<div class=\"card\">\n\n<h3>Mission CPGE<\/h3>\n\n<p>On consid\u00e8re :<\/p>\n\n<div class=\"formula\">\nA(\u03bb)=\n[[1,1,1],\n [1,\u03bb,1],\n [1,1,\u03bb]]\n<\/div>\n\n<p>Votre mission :<\/p>\n\n<ol>\n<li>calculer det(A(\u03bb)) ;<\/li>\n<li>simplifier le calcul par des op\u00e9rations sur les lignes ;<\/li>\n<li>d\u00e9terminer les valeurs de \u03bb pour lesquelles A(\u03bb) n&#8217;est pas inversible ;<\/li>\n<li>interpr\u00e9ter ces valeurs en termes de d\u00e9pendance lin\u00e9aire des colonnes ;<\/li>\n<li>\u00e9tudier le rang dans les cas particuliers ;<\/li>\n<li>comparer avec le cas g\u00e9n\u00e9rique.<\/li>\n<\/ol>\n\n<div class=\"actions\">\n<button class=\"btn gold\" type=\"button\" onclick=\"mzaDetHint('detMission1')\">\ud83d\udca1 INDICE<\/button>\n<button class=\"btn\" style=\"background:#06172c;color:#fff\" type=\"button\" onclick=\"mzaDetHint('detMission2')\">\ud83e\udde0 M\u00c9THODE<\/button>\n<\/div>\n\n<div id=\"detMission1\" class=\"hint\">\nSoustrayez la premi\u00e8re ligne aux deux autres avant de d\u00e9velopper.\n<\/div>\n\n<div id=\"detMission2\" class=\"hint\">\nEssayez de faire appara\u00eetre plusieurs z\u00e9ros. Le d\u00e9terminant devient alors beaucoup plus simple et la d\u00e9pendance en \u03bb devient imm\u00e9diatement visible.\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section class=\"dark\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\" style=\"color:#efd68c\">MZA DET LAB<\/span>\n<h2>Ne d\u00e9veloppez jamais trop t\u00f4t.<\/h2>\n\n<p>\nUn d\u00e9terminant doit d&#8217;abord \u00eatre simplifi\u00e9 par sa structure et les op\u00e9rations \u00e9l\u00e9mentaires.\n<\/p>\n<\/div>\n\n<div class=\"method\">\n<div><b>A<\/b><span>TRIANGULAIRE ?<\/span><\/div>\n<div><b>B<\/b><span>Z\u00c9ROS ?<\/span><\/div>\n<div><b>C<\/b><span>LIGNES LI\u00c9ES ?<\/span><\/div>\n<div><b>D<\/b><span>FACTORISER<\/span><\/div>\n<div><b>E<\/b><span>D\u00c9VELOPPER<\/span><\/div>\n<div><b>F<\/b><span>CONCLURE<\/span><\/div>\n<\/div>\n\n<\/div>\n<\/section>\n\n<section id=\"qcm\">\n<div class=\"wrap\">\n\n<div class=\"head\">\n<span class=\"kicker\">11. QCM DE VALIDATION<\/span>\n<h2>Diagnostic D\u00e9terminants CPGE.<\/h2>\n\n<p>Le score est affich\u00e9 sans r\u00e9v\u00e9ler les r\u00e9ponses correctes.<\/p>\n<\/div>\n\n<div class=\"qcm\">\n\n<form id=\"detQuiz\">\n\n<div class=\"question\">\n<strong>1. Pour A=[[a,b],[c,d]], det(A) vaut :<\/strong>\n<label><input type=\"radio\" name=\"d1\" value=\"1\"> ad-bc<\/label>\n<label><input type=\"radio\" name=\"d1\" value=\"0\"> ac-bd<\/label>\n<label><input type=\"radio\" name=\"d1\" value=\"0\"> ab+cd<\/label>\n<label><input type=\"radio\" name=\"d1\" value=\"0\"> a+b+c+d<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>2. Une matrice carr\u00e9e A est inversible si et seulement si :<\/strong>\n<label><input type=\"radio\" name=\"d2\" value=\"1\"> det(A)\u22600<\/label>\n<label><input type=\"radio\" name=\"d2\" value=\"0\"> det(A)=0<\/label>\n<label><input type=\"radio\" name=\"d2\" value=\"0\"> det(A)=1 uniquement<\/label>\n<label><input type=\"radio\" name=\"d2\" value=\"0\"> tous ses coefficients sont non nuls<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>3. det(AB) est \u00e9gal \u00e0 :<\/strong>\n<label><input type=\"radio\" name=\"d3\" value=\"1\"> det(A)det(B)<\/label>\n<label><input type=\"radio\" name=\"d3\" value=\"0\"> det(A)+det(B)<\/label>\n<label><input type=\"radio\" name=\"d3\" value=\"0\"> det(A)-det(B)<\/label>\n<label><input type=\"radio\" name=\"d3\" value=\"0\"> 0 toujours<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>4. \u00c9changer deux lignes d&#8217;une matrice :<\/strong>\n<label><input type=\"radio\" name=\"d4\" value=\"1\"> change le signe du d\u00e9terminant<\/label>\n<label><input type=\"radio\" name=\"d4\" value=\"0\"> ne change jamais le d\u00e9terminant<\/label>\n<label><input type=\"radio\" name=\"d4\" value=\"0\"> rend toujours le d\u00e9terminant nul<\/label>\n<label><input type=\"radio\" name=\"d4\" value=\"0\"> multiplie le d\u00e9terminant par 2<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>5. Pour une matrice triangulaire :<\/strong>\n<label><input type=\"radio\" name=\"d5\" value=\"1\"> le d\u00e9terminant est le produit des termes diagonaux<\/label>\n<label><input type=\"radio\" name=\"d5\" value=\"0\"> le d\u00e9terminant vaut toujours 1<\/label>\n<label><input type=\"radio\" name=\"d5\" value=\"0\"> le d\u00e9terminant est la somme des colonnes<\/label>\n<label><input type=\"radio\" name=\"d5\" value=\"0\"> elle n&#8217;est jamais inversible<\/label>\n<\/div>\n\n<div class=\"question\">\n<strong>6. Si det(A)=0 :<\/strong>\n<label><input type=\"radio\" name=\"d6\" value=\"1\"> les colonnes de A sont lin\u00e9airement d\u00e9pendantes<\/label>\n<label><input type=\"radio\" name=\"d6\" value=\"0\"> A est n\u00e9cessairement l&#8217;identit\u00e9<\/label>\n<label><input type=\"radio\" name=\"d6\" value=\"0\"> A est forc\u00e9ment diagonale<\/label>\n<label><input type=\"radio\" name=\"d6\" value=\"0\"> A poss\u00e8de toujours un inverse<\/label>\n<\/div>\n\n<button type=\"button\" class=\"btn gold\" onclick=\"mzaDetScore()\">\nVALIDER MON QCM\n<\/button>\n\n<\/form>\n\n<div id=\"detResult\"><\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<section>\n<div class=\"wrap\">\n\n<div class=\"cta\">\n\n<div style=\"font-size:43px\">det(A)<\/div>\n\n<h2>Un seul nombre peut r\u00e9v\u00e9ler toute la structure.<\/h2>\n\n<p>\nLe d\u00e9terminant permet de tester l&#8217;inversibilit\u00e9, l&#8217;ind\u00e9pendance,\nle changement de volume et certaines propri\u00e9t\u00e9s fondamentales\nd&#8217;un endomorphisme sans r\u00e9soudre enti\u00e8rement le probl\u00e8me.\n<\/p>\n\n<div class=\"actions\" style=\"justify-content:center\">\n<a href=\"#cours\" class=\"btn gold\">\ud83d\udcda REVOIR LE COURS<\/a>\n<a href=\"#exercices\" class=\"btn glass\">\ud83e\udde9 EXERCICES<\/a>\n<a href=\"#qcm\" class=\"btn glass\">\u2753 REFAIRE LE QCM<\/a>\n<\/div>\n\n<\/div>\n<\/div>\n<\/section>\n\n<script>\nfunction mzaDetHint(id){\n const el=document.getElementById(id);\n if(!el)return;\n el.style.display = el.style.display===\"block\" ? \"none\" : \"block\";\n}\n\nfunction mzaDetScore(){\n const form=document.getElementById(\"detQuiz\");\n const result=document.getElementById(\"detResult\");\n\n let score=0;\n let complete=true;\n\n [\"d1\",\"d2\",\"d3\",\"d4\",\"d5\",\"d6\"].forEach(function(name){\n   const answer=form.querySelector('input[name=\"'+name+'\"]:checked');\n\n   if(!answer){\n     complete=false;\n   }else{\n     score+=Number(answer.value);\n   }\n });\n\n result.style.display=\"block\";\n\n if(!complete){\n   result.innerHTML=\"<strong>R\u00e9pondez aux 6 questions avant de valider.<\/strong>\";\n   return;\n }\n\n const pct=Math.round(score\/6*100);\n\n let level=\"D\u00e9terminants \u00e0 consolider\";\n if(pct>=50) level=\"Bases acquises\";\n if(pct>=67) level=\"Bon niveau\";\n if(pct>=84) level=\"Tr\u00e8s bonne ma\u00eetrise\";\n if(pct===100) level=\"Excellent niveau CPGE\";\n\n result.innerHTML=\n \"<strong style='font-size:30px;color:#efd68c'>\"+pct+\"%<\/strong>\"+\n \"<p><b>\"+level+\"<\/b><\/p>\"+\n \"<p>Score : \"+score+\" \/ 6<\/p>\"+\n \"<p>Les r\u00e9ponses correctes et la correction d\u00e9taill\u00e9e restent masqu\u00e9es.<\/p>\";\n}\n<\/script>\n\n<\/div>\n&#8220;`\n","protected":false},"excerpt":{"rendered":"<p>&#8220;`html id=&#8221;mza-determinants-13&#8243; COURS 13 \u2022 D\u00c9TERMINANTS Mesurer.D\u00e9cider. Inverser. Le d\u00e9terminant permet de d\u00e9tecter imm\u00e9diatement l&#8217;inversibilit\u00e9 d&#8217;une matrice, mesurer un facteur de volume, tester l&#8217;ind\u00e9pendance de vecteurs et simplifier de nombreuses questions d&#8217;alg\u00e8bre lin\u00e9aire. \ud83d\udcda COMMENCER \ud83e\udde0 M\u00c9THODE \ud83e\udde9 EXERCICES \u2753 QCM 1. D\u00c9TERMINANT EN DIMENSION 2 Le premier calcul fondamental. Matrice 2\u00d72 A = [[a,b],[c,d]] [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1552","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1552","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/comments?post=1552"}],"version-history":[{"count":1,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1552\/revisions"}],"predecessor-version":[{"id":1805,"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/pages\/1552\/revisions\/1805"}],"wp:attachment":[{"href":"https:\/\/maroczain.com\/scolaire.maroczain.com\/wp-json\/wp\/v2\/media?parent=1552"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}