La trace d'une matrice : concepts et applications

Question 1 sur 10 10:00

[{"id":54871,"question":"Quelle est la trace de la matrice A = [[2, 3], [1, 4]] ?","option_a":"6","option_b":"5","option_c":"7","option_d":"4","option_e":"","option_f":"","bonne_reponse":"A","explication":"La trace est la somme des éléments diagonaux : 2 + 4 = 6.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54872,"question":"La trace d'une matrice est-elle toujours un nombre réel ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"La trace est la somme de nombres réels (les éléments diagonaux), donc elle est toujours réelle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54873,"question":"Si A et B sont deux matrices carrées de même taille, que vaut Tr(A + B) ?","option_a":"Tr(A) + Tr(B)","option_b":"Tr(A) * Tr(B)","option_c":"Tr(AB)","option_d":"Aucune des réponses","option_e":"","option_f":"","bonne_reponse":"A","explication":"La trace est une application linéaire : Tr(A + B) = Tr(A) + Tr(B).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54874,"question":"Pour une matrice triangulaire supérieure, la trace est-elle égale à la somme des éléments diagonaux ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car les éléments non diagonaux (au-dessus de la diagonale) n'interviennent pas dans le calcul de la trace.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54875,"question":"Si λ est une valeur propre de A, que vaut Tr(A) ?","option_a":"λ","option_b":"nλ (où n est la taille de A)","option_c":"La somme des valeurs propres de A","option_d":"Le produit des valeurs propres de A","option_e":"","option_f":"","bonne_reponse":"C","explication":"La trace est égale à la somme des valeurs propres de A (en comptant les multiplicités).","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54876,"question":"La trace d'une matrice est-elle invariante par changement de base ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, la trace est un invariant de similitude : Tr(P⁻¹AP) = Tr(A) pour toute matrice inversible P.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54877,"question":"Quelle est la trace de la matrice identité de taille n ?","option_a":"0","option_b":"1","option_c":"n","option_d":"n²","option_e":"","option_f":"","bonne_reponse":"C","explication":"La matrice identité a des 1 sur sa diagonale et des 0 ailleurs. Sa trace est donc n.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54878,"question":"Si Tr(A) = 0, peut-on conclure que A est la matrice nulle ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Non, par exemple la matrice [[1, -1], [1, -1]] a une trace nulle mais n'est pas la matrice nulle.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54879,"question":"Pour deux matrices A et B de même taille, Tr(AB) est-il égal à Tr(BA) ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"A","explication":"Oui, car Tr(AB) = Tr(BA) grâce à la cyclicité de la trace.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0},{"id":54880,"question":"Si A est une matrice symétrique, sa trace est-elle toujours positive ?","option_a":"Vrai","option_b":"Faux","option_c":"","option_d":"","option_e":"","option_f":"","bonne_reponse":"B","explication":"Non, par exemple la matrice [[-1, 0], [0, -1]] est symétrique mais sa trace est -2.","points":1,"type":"qcm","actif":1,"section_id":null,"ordre":0}]