Matrix Inversion

Definition

The inverse of an n by n matrix A is an n by n matrix B where,

AB = BA = In.

Please note: Not all matrices have inverses! Singular matrices don’t have inverse.

If a matrix has an inverse, then it is called invertible.

Definition

If A is a square matrix, and if a matrix B of the same size can be found such that AB=BA=I , then A is said to be invertible and B is called an inverse of A . If no such matrix B can be found, then A is said to be singular .

Notation:

1AB

Properties of Matrix Inverse

If A is an invertible matrix then its inverse is unique. (A-1)-1 = A. (Ak)-1= (A-1)k (we will denote this as A-k ) (cA)-1 = (1/c)A-1, c ≠ 0. ( AT)-1 = (A-1)T. If A is an invertible matrix, then the system of

equations Ax = b has a unique solution given by x = A-1b.

The Transpose of a Matrix A is the m×n matrix given by

The transpose matrix of A, denoted by AT, is a n×m matrix given by

.

21

22221

11211

mnmm

n

n

aaa

aaa

aaa

A

,

21

22212

12111

mnnn

m

m

T

aaa

aaa

aaa

A

The inverse of a 2-by-2 matrix

For a 2-by-2 matrix,

dc

baA

ac

bd

bcadA

11

The inverse of a 2 x 2 matrix

From this we deduced that a 2x2 matrix A is singular if and only if ad-bc = 0.

This quantity (ad-bc) has some other useful properties as well and so is defined to be the determinant of the matrix A.

Determinants of larger matrices

There is no “nice” formula for the inverse of larger than 2x2 matrices.

We still can define the determinant of a larger square matrix and it will still have the property that the determinant of A= 0 if and only if A is singular.

First we need some terminology.

Minors and cofactors

If A is a square matrix, then the minor Mij of the element aij of A is the determinant of the matrix obtained by deleting the i-th row and the j-th column from A.

The cofactor Cij = (-1)i+jMij.

Definition of a Determinant

If A is a square matrix of order 2 or greater, then the determinant of A is the sum of the entries in the first row of A multiplied by their cofactors. That is,

n

jjjCaAA

111)det(

Determinant of 2-by-2 Matrix

dc

baA

bcadAA )det(

Determinant of 3-by-3 Matrix

ihg

fed

cba

A

)()(

)det(

idbhfageccdhbfgaei

cegbfgcdhbdiafhaei

hg

edc

ig

fdb

ih

feaAA

Matrix Inversion

How to calculate the matrix inverse?

What is an adjoint matrix?

)(11 AadjA

A

Adjoint Matrix – Minors and Cofactors

The adjoint matrix of [A], Adj[A] is obtained by taking the transpose of the cofactor matrix of [A].

The minor for element aij of matrix [A] is found by removing the ith row and jth column from [A] and then calculating the determinant of the remaining matrix.

Matrix Inversion

1 2 3

1 2 3

1 2 3

2 4 5 36

-3 5 7 7

5 3 8 -31

x x x

x x x

x x x

Consider the following set of linear equations.

The coefficients can be arranged in a matrix form as shown.

2 -4 5

-3 5 7

5 3 -8

A

Matrix Inversion

1

2

3

2 -4 5 36

-3 5 7 7

5 3 -8 -31

x

x

x

[ ][ ] [ ]A x B=

The set of equations in matrix form is:

[ ] [ ] [ ]1x A B-=

Minors

835

753

542

835

753

542

835

753

542

11

5 7-61

3 -8M = =

12

-3 7-11

5 -8M = =

13

-3 5-34

5 3M = =

Minors

-61 -11 -34

17 -41 26

-53 29 -2

M

23

2 -426

5 3M

The resulting matrix of minors is:

835

753

542

Cofactors

Cofactors are the signed minors. The cofactor of element aij of matrix [A] is:

Therefore

The resulting matrix of cofactors is:

( )1 3

13 13-1C M+=

( )-1 i j

ij ijC M+=

( )1 2

12 12-1C M+=

( )1 1

11 11-1C M+=

-61 11 -34

-17 -41 -26

-53 -29 -2

C

Adjoint matrixThe adjoint matrix of [A], Adj[A] is obtained by taking the transpose of the cofactor matrix of [A].

-61 11 -34

-17 -41 -26

-53 -29 -2

C

Evaluate the determinant

-61 -17 -53

11 -41 -29

-34 -26 -2

adj A

2 -4 5

-3 5 7 -336

5 3 -8

A = =

[ ] [ ]Tadj A C=

Matrix Inversion [ ] [ ]1 1A adj A

A- =

1

-61 -17 -53

-336 -336 -33611 -41 -29

-336 -336 -336-34 -26 -2

-336 -336 -336

A

1

-61 -17 -531

11 -41 -29-336

-34 -26 -2

A

1

61 17 53

336 336 336-11 41 29

336 336 33617 13 1

168 168 168

A

Matrix Inversion Using G-J Elimination

If Gauss–Jordan elimination is applied on a square matrix, it can be used to calculate the matrix's inverse. This can be done by augmenting the square matrix with the identity matrix of the same dimensions., and through the following matrix operations:

IA IAA 1 1AI

210

121

012

A

If the original square matrix, A, is given by the following expression:

Then, after augmenting by the identity, the following is obtained:

100

010

001

210

121

012

IA

4

3

2

1

4

12

11

2

14

1

2

1

4

3

100

010

0011AI

100

010

001

I

By performing elementary row operations on the [AI] matrix until A reaches reduced row echelon form, the following is the final result:

The matrix augmentation can now be undone, which gives the following:

4

3

2

1

4

12

11

2

14

1

2

1

4

3

1A