[PDF] [PDF] Inverse dune matrice carrée

[PDF] Cours 3: Inversion des matrices dans la pratique - Institut de

Rappel de l'épisode précédent sur l'inverse d'une application linéaire/matrice Pivot de Gauss Module complémentaire de maths, année 2012 Clément Rau

[PDF] Inverse dune matrice carrée

Nous nous intéressons ici aux matrices carrées (autant de lignes que de colonnes) en vue de la résolution de Ax = b (autant d'équations que d'inconnues )

[PDF] Exercices Corrigés Matrices Exercice 1 – Considérons les matrices

Exercice 12 – Soit A et B deux matrices carrées de même ordre, on suppose que la matrice AB est inversible d'inverse la matrice C Montrer alors que B est 

[PDF] FONCTION INVERSE - maths et tiques

Yvan Monka – Académie de Strasbourg – www maths-et-tiques FONCTION INVERSE I Définition et allure de la courbe Vidéo https://youtu be/Vl2rlbFF22Y

[PDF] Utiliser linverse dune matrice pour résoudre un - Lycée dAdultes

Utiliser l'inverse d'une matrice pour résoudre un système d'équations courbes polynomiales Exercice 1 : Dans une ferme, il y a des lapins et des poules

[PDF] Matrices - Exo7 - Cours de mathématiques

C'est une matrice inversible, et son inverse est elle-même par l'égalité InIn = In • La matrice nulle 0n de taille n × n n'est pas inversible En effet on sait que, pour 

[PDF] Inversion de matrices - FOAD - MOOC

Math Stat 1 Module 3 : Inversion de matrices M3 L3_MS1_M3 1/5 Module 3 : Inversion de matrices Unité 1 Définition On ne définira l'inverse d'une matrice 

[PDF] Ch7 : Division de fractions 1 Inverse 2 Propriétés des inverses 3

L'inverse d'un nombre x est le nombre noté x−1 par lequel le multiplier pour obtenir 1, c'est à dire pour obtenir x × x−1 = 1 Théorème L'inverse de x non nul  

[PDF] Introduction aux problèmes inverses - ORBi

C Charles∗ RÉSUMÉ Cette note technique est une initiation aux problèmes inverses of Inverse Problems Number 120 in Applied Mathematical Sciences

[PDF] cours d anthropologie

[PDF] philosophie africaine selon hegel pdf

[PDF] cours et exercices de mathematique financiere pdf

[PDF] le capitaine dit a son fils la cabine n°1

[PDF] dossier sur l’autonomie

[PDF] jdi n°9

[PDF] mai 1993

[PDF] travail de groupe définition

[PDF] demander des informations sur un voyage organisé

[PDF] qu est ce que la geologie

[PDF] les branches de la geologie pdf

[PDF] importance de la geomorphologie

[PDF] definition de la geologie

[PDF] definition de la geologie generale

[PDF] united nations charter pdf

Matrix inversion

Math 130 Linear Algebra

D Joyce, Fall 2015

We'll start o with the denition of the inverse

of a square matrix and a couple of theorems.

Denition 1.We say that two squarennma-

tricesAandBareinversesof each other if

AB=BA=I

and in that case we say thatBis an inverse ofA and thatAis an inverse ofB. If a matrix has no inverse, it is said to besingular, but if it does have an inverse, it is said to beinvertibleornonsingular.

Theorem 2.A matrixAcan have at most one

inverse. The inverse of an invertible matrix is denotedA1. Also, when a matrix is invertible, so is its inverse, and its inverse's inverse is itself, (A1)1=A.

Proof.Suppose thatBandCare both inverses of

A. Then bothAB=BA=IandAC=CA=I.

Therefore

B=BI=B(AC) = (BA)C=IC=C

Thus, there is at most one inverse.

The second statement (A1)1=Afollows from

the denition of the inverse ofA1, namely, its in- verse is the matrixBsuch thatA1B=BA1=I.

SinceAhas that property, thereforeAis the inverse

ofA1.q.e.d.

Theorem 3.IfAandBare both invertible, then

their product is, too, and (AB)1=B1A1.

Proof.Since there is at most one inverse ofAB,

all we have to show is thatB1A1has the prop- erty required to be an inverse ofAB, name, that (AB)(B1A1) = (B1A1)(AB) =I. But that follows from associativity of matrix multiplication and the facts thatAA1=A1A=IandBB1= B

1B=I.q.e.d.Inverses of22matrices.You can easily nd

the inverse of a 22 matrix. Consider a generic

22 matrix

A=a b c d

It's inverse is the matrix

A

1=d=b=

c=a= where is the determinant ofA, namely =adbc; provided is not 0. In words, to nd the inverse of a 22 matrix, (1) exchange the entries on the major diagonal, (2) negate the entries on the mi- nor diagonal, and (3) divide all four entries by the determinant.

It's easy to verify thatA1actually is the inverse

ofA, just multiply them together to get the identity matrixI.

A method for nding inverse matrices.Next

we'll look at a dierent method to determine if an nnsquare matrixAis invertible, and if it is what it's inverse is.

The method is this. First, adjoin the identity

matrix to its right to get ann2nmatrix [AjI]. Next, convert that matrix to reduced echelon form. If the result looks like [IjB], thenBis the desired inverseA1. But if the square matrix in the left half of the reduced echelon form is not the identity, thenAhas no inverse.

We'll verify that this method works later.

Example 4.Let's illustrate the method with a 3

3 example. LetAbe the matrix

A=2 432 4
1 0 2

0 1 03

5

Form the 36 matrix [AjI], and row reduce it.

I'll use the symbolwhen a row-operation is ap-

1 plied. Here are the steps. [AjI] =2

432 41 0 0

1 0 20 1 0

0 1 00 0 1

3 5 2

41 0 20 1 0

0 1 00 0 1

32 41 0 0

3 5 2

41 0 20 1 0

0 1 00 0 1

02213 03

5 2

41 0 20 1 0

0 1 00 0 1

0 0213 23

5 2

41 0 012 2

0 1 00 0 1

0 0 11=2 3=213

5 = [IjA1]

This row-reduction to reduced echelon form suc-

ceeded in turning the left half of the matrix into the identity matrix. When that happens, the right half of the matrix will be the inverse matrixA1.

Therefore, the inverse matrix is

A 1=2 412 2
0 0 1

1=2 3=213

5

Matlabcan compute inversesor tell you if

they're singular. >> A = [1 2; 3 4] A = 1 2 3 4 >> B = inv(A) B = -2.0000 1.0000

1.5000 -0.5000

>> A*B ans =

1.0000 00.0000 1.0000

>> C = [1 2; 3 6] C = 1 2 3 6 >> D = inv(C)

Warning: Matrix is singular to working precision.

D =

Inf Inf

Inf Inf

Math 130 Home Page at

http://math.clarku.edu/ ~ma130/ 2quotesdbs_dbs6.pdfusesText_11