Montrons seulement que la condition est suffisante

  • No School
  • AA 1
  • 34

This preview shows page 24 - 34 out of 34 pages.

Montrons seulement que la condition est suffisante : soient donc X et Y deux vecteurs de R n . Si on a Y = AX , en multipliant (` a gauche !) par B , on obtient BY = X . R´ eciproquement, si on a X = BY , on obtient AX = Y en multipliant par A (toujours ` a gauche).
Image of page 24

Subscribe to view the full document.

Calcul d’inverse : exemple Exemple Calculons l’inverse de la matrice 1 2 3 5 . Pour cela, on calcule la r´ eciproque de l’application lin´ eaire associ´ ee en r´ esolvant le syst` eme x + 2 y = x 3 x + 5 y = y . Par combinaison lin´ eaire, on trouve x = - 5 x + 2 y y = 3 x - y . L’inverse cherch´ e est donc - 5 2 3 - 1 .
Image of page 25
Calcul d’inverse : slogan Slogan Pour calculer l’inverse de la matrice A , on r´ esout le syst` eme AX = X , o`u X est le vecteur inconnu.
Image of page 26

Subscribe to view the full document.

Calcul d’inverse : exercice Exo 4 Calculez l’inverse de la matrice 1 2 2 3 .
Image of page 27
Calcul d’inverse : exemple Exemple Soit A une matrice carr´ ee de taille 2 v´ erifiant A 2 - 2 A - 3 I = 0 . On a A 2 - 2 A = A ( A - 2 I ) = 3 I et donc A - 1 = 1 3 ( A - 2 I ) .
Image of page 28

Subscribe to view the full document.

Calcul d’inverse : exercice Exo 5 Calculez l’inverse de la matrice A := 1 2 3 4 sachant qu’elle erifie A 2 - 5 A - 2 I = 0 .
Image of page 29
Crit` ere d’inversibilit´ e : exemple On a compris que la matrice A := 5 2 3 4 est inversible quand le syst` eme 5 x + 2 y = x 3 x + 4 y = y . aux inconnues x et y a une unique solution. C’est le cas exactement quand la matrice A est de rang 2. Et c’est en´ eral.
Image of page 30

Subscribe to view the full document.

Crit` ere d’inversibilit´ e : cas g´ en´ eral Exo 6 Calculez l’inverse de la matrice A := 1 2 3 4 sachant qu’elle erifie A 2 - 5 A - 2 I = 0 . Proposition Une matrice carr´ ee de taille n est inversible ssi son rang est n .
Image of page 31
Crit` ere d’inversibilit´ e : exercice Exo 7 ecidez si la matrice A := 1 2 3 5 4 5 5 - 2 - 5 est inversible.
Image of page 32

Subscribe to view the full document.

Interpolation Probl` eme Montrer qu’il existe un unique trinˆ ome du second degr´ e P erifiant : P (1) = 4 , P (2) = 1 , P (5) = 7 .
Image of page 33
Image of page 34
  • Fall '19
  • Nombre, Matrice, produit matriciel, Mq,r, Mp,r

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes