## 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  - 1 [-2] -1 [4+2]

= 6 +2 -1 

= 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

 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 

= 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

 -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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