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 nonsingular matrix,there exists an inverse which is given by
Example 1:
Find the inverse of the following matrix

A = 3 [20]  1 [20] 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 


= [20] = 2 
minor of 1 


= [20] = 2 
minor of 1 


= [4(2)] = [4+2] = 6 
minor of 2 


= [1(2)] = [1+2] = 1 
minor of 2 


= [3(1)] = [3+1] = 2 
minor of 0 


= [61] = 5 
minor of 1 


= [02] = 2 
minor of 2 


= [0(2)] = 2 
minor of 1 


= [62] = 8 
minor matrix = 


cofactor matrix = 


Adjoint matrix = 

A⁻¹ =1/2 

Example 2:
Find the inverse of the following matrix

A = 3 [6(4)]  4 [010] +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 


= [6(4)] = (6+4) = 2 
minor of 4 


= [010] = (10) = 10 
minor of 1 


= [0(5)] = [0+5] = 5 
minor of 0 


= [24(2)] = [24+2] = 26 
minor of 1 


= [185] = 13 
minor of 2 


= [620] = 26 
minor of 5 


= [8(1)] = (8+1) = 9 
minor of 2 


= [60] = 6 
minor of 6 


= [30] = 3 
minor matrix 


Cofactor matrix 


Adjoint of matrix (adj A) 

A⁻¹=1/39 

These are the examples in the page inverse of a matrix.
Questions 
Solution  
1) Find the inverse of the following matrix 


2) Find the inverse of the following matrix 


3) Find the inverse of the following matrix 


4) Find the inverse of the following matrix 


5) Find the inverse of the following matrix 


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