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:
B  A1
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
 a11 a12  a1n 
a
a22  a2 n 
21
.
A

 
 


a
a

a
mn 
 m1 m 2
 a11 a21  am1 
a a  a 
m2 
T
A   12 22
,

 



a
a

a
mn 
 1n 2 n
The inverse of a 2-by-2 matrix

For a 2-by-2 matrix,
a b 
A

c d 
1  d  b
A 



c
a
ad  bc 

1
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
det (A)  A   a1 j C1 j
j 1
Determinant of 2-by-2 Matrix
a
A
c
b
d 
det(A)  A  ad  bc
Determinant of 3-by-3 Matrix
a
A   d
 g
e
det (A)  A  a
h
b
e
h
c
f 
i 
f
d
b
i
g
f
d
c
i
g
e
h
 aei  afh  bdi  cdh  bfg  ceg
 ( aei  bfg  cdh)  ( gec  hfa  idb)
Matrix Inversion
How to calculate the matrix inverse?
A
1
1

adj( A)
A
What is an adjoint matrix?
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
Consider the following set of linear equations.
2 x1  4 x2  5 x3  36

-3 x1  5 x2  7 x3  7
5 x  3 x  8 x  -31
2
3
 1
The coefficients can be arranged in a matrix form as shown.
 2 -4 5 
 -3 5 7 
A

  

 5 3 -8 
Matrix Inversion
The set of equations in matrix form is:
 2 -4 5   x1   36 
-3 5 7   x    7 

 2  
 5 3 -8  x3   -31
[A ][x ]= [B ]
[x ]=
- 1
[A ] [B ]
Minors
 2
 3

 5
4
 2
 3

 5
4
 2
 3

 5
4
5
3
5
3
5
3
5 
7 
 8
5 
7 
 8
5 
7 
 8
5
M 11 =
3 -8
-3
M 12 =
7
5
= -61
7
-8
= -11
-3 5
M 13 =
5
3
= -34
Minors
 2 4 5 
 3 5

7


 5
3  8
M 23
2 -4

 26
5 3
The resulting matrix of minors is:
 -61 -11 -34 
 M    17 -41 26 
 -53 29 -2 
Cofactors
Cofactors are the signed minors.
The cofactor of element aij of matrix [A] is: C
Therefore
1+ 1
M 11
1+ 2
M 12
1+ 3
M 13
C 11 = (-1)
C 12 = (-1)
C 13 = (-1)
The resulting matrix of cofactors is:
i+ j
ij
= (-1)
 -61 11 -34 
C   -17 -41 -26 
 -53 -29 -2 
M ij
Adjoint matrix
The adjoint matrix of [A], Adj[A] is obtained by taking the
T
adj
A
=
C
[ ] [ ]
transpose of the cofactor matrix of [A].
 -61 11 -34 
C   -17 -41 -26 
 -53 -29 -2 
Evaluate the determinant
 -61 -17 -53 
adj  A   11 -41 -29 
 -34 -26 -2 
2
-4
5
A = -3
5
7 = -336
5
3
-8
Matrix Inversion
 A
1
 A
1
 -61
 -336

11

 -336

 -34
 -336
-17
-336
-41
-336
-26
-336
-53 
-336 

-29 
-336 

-2 
-336 
1
adj [A ]
[A ] =
A
- 1
 -61 -17 -53 
1 


11
-41
-29

-336 
 -34 -26 -2 
 A
1
 61
 336

-11

 336

 17
 168
17
336
41
336
13
168
53 
336 

29 
336 

1 
168 
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:
A
I
1
A
A
I
I
1
A

If the original square matrix, A, is given by the following expression:
2
A   1

 0
1
2
1
0
 1

2 
Then, after augmenting by the
identity, the following is obtained:
A
2
I    1

 0
1
0
1
0
2
1
1
2
0
0
1
0
0
0

1
By performing elementary row operations on the [AI] matrix until
A reaches reduced row echelon form, the following is the final result:
I
A1


1

 0
0


0
1
0
0
0
1
3
4
1
2
1
4
1
2
1
1
2
1
4
1

2
3
4 
The matrix augmentation can now be undone, which gives the following:
1
I  0

0
0
1
0
0
0

1
A1
3
4
1

2
1
 4
1
2
1
1
2
1
4
1

2
3
4 