Quiz — 2-Suites et Séries de Matrices , Exponentielle d_une matrice Correction.pdf

Question 1 sur 10 20:00

[{"id":34640,"question":"Soit A une matrice carrée. Quelle est la formule de l'exponentielle de A ?","option_a":"exp(A) = A^A","option_b":"exp(A) = Σ (A^n)\/n! pour n de 0 à ∞","option_c":"exp(A) = det(A) * I","option_d":"exp(A) = A + I","option_e":"","option_f":"","bonne_reponse":"b","explication":"L'exponentielle d'une matrice A est définie par la série entière exp(A) = Σ (A^n)\/n! pour n de 0 à ∞, où A^0 = I (matrice identité).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"b\", \"options\": {\"a\": \"exp(A) = A^A\", \"b\": \"exp(A) = Σ (A^n)\/n! pour n de 0 à ∞\", \"c","_debug_options_count":4},{"id":34641,"question":"Une suite de matrices (A_n) converge vers une matrice L si et seulement si chaque coefficient de A_n converge vers le coefficient correspondant de L.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : la convergence d'une suite de matrices se définit coefficient par coefficient.","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":34642,"question":"Quelle propriété vérifie l'exponentielle d'une matrice A ?","option_a":"exp(A) = exp(-A)","option_b":"exp(A + B) = exp(A) + exp(B)","option_c":"exp(A) * exp(B) = exp(A + B)","option_d":"exp(A) = A^2","option_e":"","option_f":"","bonne_reponse":"c","explication":"L'exponentielle vérifie exp(A) * exp(B) = exp(A + B) lorsque A et B commutent (AB = BA).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"exp(A) = exp(-A)\", \"b\": \"exp(A + B) = exp(A) + exp(B)\", \"c\": \"exp","_debug_options_count":4},{"id":34643,"question":"Soit A une matrice nilpotente d'ordre 2 (A² = 0). Que vaut exp(A) ?","option_a":"exp(A) = I + A","option_b":"exp(A) = I + A + A²\/2","option_c":"exp(A) = I","option_d":"exp(A) = A","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une matrice nilpotente d'ordre 2, la série s'arrête à A²\/2! = 0, donc exp(A) = I + A.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"a\", \"options\": {\"a\": \"exp(A) = I + A\", \"b\": \"exp(A) = I + A + A²\/2\", \"c\": \"exp(A) = I\"","_debug_options_count":4},{"id":34644,"question":"La série Σ (A^n) converge si et seulement si le rayon spectral de A est strictement inférieur à 1.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : la convergence de la série Σ (A^n) dépend du rayon spectral de A.","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":34645,"question":"Quelle est la dérivée de exp(tA) par rapport à t ?","option_a":"d\/dt exp(tA) = A * exp(tA)","option_b":"d\/dt exp(tA) = t * exp(tA)","option_c":"d\/dt exp(tA) = exp(tA) * A","option_d":"d\/dt exp(tA) = A^t","option_e":"","option_f":"","bonne_reponse":"c","explication":"La dérivée de exp(tA) par rapport à t est A * exp(tA) = exp(tA) * A (car A et exp(tA) commutent).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"d\/dt exp(tA) = A * exp(tA)\", \"b\": \"d\/dt exp(tA) = t * exp(tA)\", \"","_debug_options_count":4},{"id":34646,"question":"Soit A une matrice diagonale de coefficients (λ₁, λ₂, ..., λₙ). Que vaut exp(A) ?","option_a":"exp(A) est la matrice nulle","option_b":"exp(A) est la matrice identité","option_c":"exp(A) est la matrice diagonale de coefficients (exp(λ₁), exp(λ₂), ..., exp(λₙ))","option_d":"exp(A) = det(A) * I","option_e":"","option_f":"","bonne_reponse":"c","explication":"Si A est diagonale, exp(A) est la matrice diagonale dont les coefficients sont les exponentielles des coefficients de A.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"c\", \"options\": {\"a\": \"exp(A) est la matrice nulle\", \"b\": \"exp(A) est la matrice identit","_debug_options_count":4},{"id":34647,"question":"La série Σ (A^n)\/n! converge toujours pour toute matrice carrée A.","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"a","explication":"C'est vrai : la série entière définissant exp(A) converge pour toute matrice carrée A.","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":34648,"question":"Quelle condition doit vérifier une matrice A pour que exp(A) soit inversible ?","option_a":"A doit être symétrique","option_b":"A doit être inversible","option_c":"A doit être nilpotente","option_d":"A peut être quelconque","option_e":"","option_f":"","bonne_reponse":"d","explication":"L'exponentielle d'une matrice est toujours inversible, car det(exp(A)) = exp(tr(A)) ≠ 0.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0,"_debug_answer_data_type":"string","_debug_answer_data_preview":"{\"correct\": \"d\", \"options\": {\"a\": \"A doit être symétrique\", \"b\": \"A doit être inversible\", \"c\": \"","_debug_options_count":4},{"id":34649,"question":"Soit A une matrice telle que A³ = 0. Que vaut exp(A) ?","option_a":"exp(A) = I + A + A²\/2","option_b":"exp(A) = I + A","option_c":"exp(A) = I","option_d":"exp(A) = A","option_e":"","option_f":"","bonne_reponse":"a","explication":"Pour une matrice nilpotente d'ordre 3, la série s'arrête à A²\/2!, donc exp(A) = I + A + A²\/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\": \"exp(A) = I + A + A²\/2\", \"b\": \"exp(A) = I + A\", \"c\": \"exp(A) = I\"","_debug_options_count":4}]