Quiz — examen_2_-mp-algebre-ipein-_2016.pdf

Question 1 sur 10 20:00

[{"id":13824,"question":"Soit A une matrice carrée d'ordre 3. Si det(A) = 0, alors :","option_a":"A est inversible","option_b":"A n'est pas inversible","option_c":"A est symétrique","option_d":"A est diagonale","option_e":"","option_f":"","bonne_reponse":"b","explication":"Un déterminant nul signifie que la matrice n'est pas inversible, car son rang est inférieur à son ordre.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"A est inversible\", \"b\": \"A n'est pas inversible\", \"c\": \"A est sym","_debug_options_count":4},{"id":13825,"question":"La matrice identité I_n est-elle toujours inversible ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"La matrice identité est inversible et son inverse est elle-même, car I_n × I_n = I_n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":13826,"question":"Quel est le déterminant d'une matrice triangulaire supérieure ?","option_a":"Le produit des éléments diagonaux","option_b":"La somme des éléments diagonaux","option_c":"Le produit des éléments hors diagonale","option_d":"Zéro","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une matrice triangulaire, le déterminant est égal au produit des éléments de la diagonale principale.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Le produit des éléments diagonaux\", \"b\": \"La somme des élémen","_debug_options_count":4},{"id":13827,"question":"Si λ est une valeur propre d'une matrice A, alors :","option_a":"det(A - λI) = 0","option_b":"det(A + λI) = 0","option_c":"tr(A) = λ","option_d":"A est inversible","option_e":"","option_f":"","bonne_reponse":"a","explication":"Par définition, λ est une valeur propre si et seulement si det(A - λI) = 0, où I est la matrice identité.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"det(A - λI) = 0\", \"b\": \"det(A + λI) = 0\", \"c\": \"tr(A) = λ\", \"d","_debug_options_count":4},{"id":13828,"question":"Une matrice symétrique est toujours diagonalisable dans ℝ.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Toute matrice symétrique réelle est diagonalisable dans ℝ, d'après le théorème spectral.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":13829,"question":"Quel est le rang d'une matrice nulle ?","option_a":"0","option_b":"1","option_c":"n (ordre de la matrice)","option_d":"Indéterminé","option_e":"","option_f":"","bonne_reponse":"a","explication":"La matrice nulle a tous ses éléments égaux à zéro, donc son rang est 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"0\", \"b\": \"1\", \"c\": \"n (ordre de la matrice)\", \"d\": \"Indéterminé","_debug_options_count":4},{"id":13830,"question":"Soit f une application linéaire de ℝ³ dans ℝ². Quelle est la dimension maximale possible de Im(f) ?","option_a":"1","option_b":"2","option_c":"3","option_d":"4","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'image d'une application linéaire de ℝ³ dans ℝ² a une dimension maximale égale à la dimension de l'espace d'arrivée, soit 2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"1\", \"b\": \"2\", \"c\": \"3\", \"d\": \"4\"}}","_debug_options_count":4},{"id":13831,"question":"Le noyau d'une application linéaire est toujours un sous-espace vectoriel.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le noyau d'une application linéaire est un sous-espace vectoriel de l'espace de départ, car il est stable par addition et multiplication par un scalaire.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4},{"id":13832,"question":"Si A est une matrice carrée d'ordre n, alors tr(A) = :","option_a":"det(A)","option_b":"somme des éléments diagonaux","option_c":"produit des valeurs propres","option_d":"A + A^T","option_e":"","option_f":"","bonne_reponse":"b","explication":"La trace d'une matrice est définie comme la somme de ses éléments diagonaux.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"det(A)\", \"b\": \"somme des éléments diagonaux\", \"c\": \"produit des","_debug_options_count":4},{"id":13833,"question":"Une matrice A est diagonalisable si et seulement si elle possède n valeurs propres distinctes (n étant son ordre).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une matrice est diagonalisable si elle possède une base de vecteurs propres, ce qui n'implique pas nécessairement que toutes ses valeurs propres soient distinctes.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"Vrai\", \"b\": \"Faux\", \"c\": \"\", \"d\": \"\"}}","_debug_options_count":4}]