Quiz — matricesexoscorriges.pdf

Question 1 sur 10 20:00

[{"id":20649,"question":"Quelle est la dimension d'une matrice résultant de la multiplication d'une matrice 3x4 par une matrice 4x2 ?","option_a":"3x2","option_b":"4x4","option_c":"2x3","option_d":"6x8","option_e":"","option_f":"","bonne_reponse":"a","explication":"Le produit de deux matrices A (m×n) et B (n×p) donne une matrice de dimension m×p. Ici, 3×4 × 4×2 = 3×2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"3x2\", \"b\": \"4x4\", \"c\": \"2x3\", \"d\": \"6x8\"}}","_debug_options_count":4},{"id":20650,"question":"Le déterminant d'une matrice triangulaire supérieure est égal au produit des éléments de sa diagonale principale.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"Cette propriété est vraie : pour une matrice triangulaire (supérieure ou inférieure), le déterminant est bien le produit des éléments diagonaux.","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":20651,"question":"Si A est une matrice carrée inversible, alors det(A⁻¹) = ?","option_a":"det(A)","option_b":"1\/det(A)","option_c":"det(A)²","option_d":"0","option_e":"","option_f":"","bonne_reponse":"b","explication":"On sait que det(A⁻¹) = 1\/det(A) car det(A × A⁻¹) = det(I) = 1 et det(A × A⁻¹) = det(A) × det(A⁻¹).","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\": \"1\/det(A)\", \"c\": \"det(A)²\", \"d\": \"0\"}}","_debug_options_count":4},{"id":20652,"question":"Quel est le rang de la matrice suivante ? \u003Cbr\u003E [[1, 2, 3], [4, 5, 6], [7, 8, 9]]","option_a":"1","option_b":"2","option_c":"3","option_d":"0","option_e":"","option_f":"","bonne_reponse":"b","explication":"Les lignes de cette matrice sont linéairement dépendantes (la 3ème ligne est la somme des 1ère et 2ème). Le rang est donc 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\": \"0\"}}","_debug_options_count":4},{"id":20653,"question":"Une matrice de rang 1 peut-elle être inversible ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Une matrice inversible doit être carrée et de rang égal à sa taille. Une matrice de rang 1 (non carrée ou carrée de taille \u003E1) n'est pas inversible.","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},{"id":20654,"question":"Quelle est la transposée de la matrice A = [[1, 2], [3, 4]] ?","option_a":"[[1, 3], [2, 4]]","option_b":"[[2, 1], [4, 3]]","option_c":"[[1, 2], [3, 4]]","option_d":"[[4, 3], [2, 1]]","option_e":"","option_f":"","bonne_reponse":"a","explication":"La transposée d'une matrice s'obtient en échangeant les lignes et les colonnes. Ainsi, Aᵀ = [[1, 3], [2, 4]].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"[[1, 3], [2, 4]]\", \"b\": \"[[2, 1], [4, 3]]\", \"c\": \"[[1, 2], [3, 4]","_debug_options_count":4},{"id":20655,"question":"Si A et B sont deux matrices carrées de même taille, alors det(A + B) = det(A) + det(B).","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Cette égalité est fausse. Le déterminant n'est pas linéaire : det(A + B) ≠ det(A) + det(B) en général.","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},{"id":20656,"question":"Quel est l'inverse de la matrice A = [[2, 1], [1, 1]] ?","option_a":"[[1, -1], [-1, 2]]","option_b":"[[1, 1], [1, 2]]","option_c":"[[-1, 1], [1, -2]]","option_d":"[[2, -1], [-1, 1]]","option_e":"","option_f":"","bonne_reponse":"a","explication":"L'inverse de A est calculé par la formule A⁻¹ = (1\/det(A)) × [[d, -b], [-c, a]] avec A = [[a, b], [c, d]]. Ici, det(A) = 1, donc A⁻¹ = [[1, -1], [-1, 2]].","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"[[1, -1], [-1, 2]]\", \"b\": \"[[1, 1], [1, 2]]\", \"c\": \"[[-1, 1], [1,","_debug_options_count":4},{"id":20657,"question":"Le produit de deux matrices symétriques est-il toujours symétrique ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"b","explication":"Non, le produit de deux matrices symétriques n'est pas nécessairement symétrique. Par exemple, [[1, 0], [0, 0]] × [[0, 0], [0, 1]] = [[0, 0], [0, 0]], qui est symétrique, mais ce n'est pas toujours le cas.","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},{"id":20658,"question":"Quelle propriété caractérise une matrice orthogonale ?","option_a":"A × Aᵀ = I","option_b":"det(A) = 1","option_c":"A est symétrique","option_d":"A est triangulaire","option_e":"","option_f":"","bonne_reponse":"a","explication":"Une matrice orthogonale vérifie A × Aᵀ = I (ou Aᵀ × A = I), ce qui signifie que son inverse est égale à sa transposée.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"A × Aᵀ = I\", \"b\": \"det(A) = 1\", \"c\": \"A est symétrique\", \"d\":","_debug_options_count":4}]