Inverse of a Matrix





In this page we are going to see how to find inverse of a matrix.

1) Reversal law for inverse

If A and B are any two non singular matrices of the same order,then AB is also non singular and (AB)⁻¹ = B⁻¹ A⁻¹ the inverse of a product is the product of the inverses taken in the reverse order.

2) Reversal law of Transposes

If A and B are matrices comfortable to multiplication,then (AB)^T = B^T A^T

3) Inverse law

For any non singular matrix A. (A^T)⁻¹ = (A⁻¹)^T

These are the properties in the topic inverse of a matrix.

Definition:

If A is a non-singular matrix,there exists an inverse which is given by

Example 1:

Find the inverse of the following matrix

 
3 1 -1
2 -2 0
1 2 -1
 

|A| = 3 [2-0] - 1 [-2-0] -1 [4-(-2)]

      = 3 [2] - 1 [-2] -1 [4+2]

      = 6 +2 -1 [6]

      = 6 +2 -6

|A| = 2 ≠ 0

Since A is a non singular matrix. A⁻¹ exists.

minor of 3

=
 
-2 0
2 -1
 

   = [2-0]

   =  2

minor of 1

=
 
2 0
1 -1
 

   = [-2-0]

   =  -2

minor of -1

=
 
2 -2
1 2
 

   = [4-(-2)]

   = [4+2]

   =  6

minor of 2

=
 
1 -1
2 -1
 

   = [-1-(-2)]

   = [-1+2]

   =  1

minor of -2

=
 
3 -1
1 -1
 

   = [-3-(-1)]

   = [-3+1]

   =  -2

minor of 0

=
 
3 1
1 2
 

   = [6-1]

   =  5

minor of 1

=
 
1 -1
-2 0
 

   = [0-2]

   =  -2

minor of 2

=
 
3 -1
2 0
 

   = [0-(-2)]

   =  2

minor of -1

=
 
3 1
2 -2
 

   = [-6-2]

   =  -8

minor matrix   =

 
2 -2 6
1 -2 5
-2 2 -8
 

cofactor matrix =

 
2 2 6
-1 -2 -5
-2 -2 -8
 

Adjoint matrix =

 
2 -1 -2
2 -2 -2
6 -5 -8
 

 A⁻¹ =1/2

 
2 -1 -2
2 -2 -2
6 -5 -8
 

Example 2:

Find the inverse of the following matrix

 
3 4 1
0 -1 2
5 -2 6
 

|A| = 3 [-6-(-4)] - 4 [0-10] +1 [0-(-5)]

      = 3 [-6+4] - 4 [-10] +1 [5]

      = 3 [-2] + 40 + 5

      = -6 + 40 + 5

      = -6 + 45

      = 39

|A| = 39 ≠ 0

Since A is a non singular matrix. A⁻¹ exists.

minor of 3

=
 
-1 2
-2 6
 

   = [-6-(-4)]

   = (-6+4)

   = -2

minor of 4

=
 
0 2
5 6
 

   =  [0-10]

   =  (-10)

   = -10

minor of 1

=
 
0 -1
5 -2
 

   =  [0-(-5)]

   =  [0+5]            

   =  5

minor of 0

=
 
4 1
-2 6
 

   = [24-(-2)]

   = [24+2]

   =  26

minor of -1

=
 
3 1
5 6
 

   = [18-5]

   =  13

minor of 2

=
 
3 4
5 -2
 

   = [-6-20]

   =  -26

minor of 5

=
 
4 1
-1 2
 

   =  [8-(-1)]

   =  (8+1)

   =  9

minor of -2

=
 
3 1
0 2
 

   =  [6-0]

   =  6

minor of 6

=
 
3 4
0 -1
 

   =  [-3-0]

   =  -3

minor matrix

 
-2 -10 5
26 13 -26
9 6 -3
 

Co-factor matrix

 
-2 10 5
-26 13 26
9 -6 -3
 

Ad-joint of matrix (adj A)

 
-2 -26 9
10 13 -6
5 26 -3
 

 A⁻¹=1/39

 
-2 -26 9
10 13 -6
5 26 -3
 

These are the examples in the page inverse of a matrix.












Questions



Solution


1) Find the inverse of the following matrix

 
2 1 1
1 1 1
1 -1 2
 

Solution

2) Find the inverse of the following matrix

 
1 2 1
2 -1 2
1 1 -2
 

Solution

3) Find the inverse of the following matrix

 
6 2 3
3 1 1
10 3 4
 

Solution

4) Find the inverse of the following matrix

 
2 5 7
1 1 1
2 1 -1
 

Solution

5) Find the inverse of the following matrix

 
3 1 -1
2 -1 2
2 1 -2
 

inverse of a matrix

Solution







Inverse of a Matrix to Minor of a Matrix
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