Inverse of a Matrix

Given a matrix A, if there exists a matrix B such that AB = BA = I, then B is called inverse of A. When we multiply a number by its reciprocal we get 1 and when we multiply a matrix by its inverse we get Identity matrix. Inverse of A is denoted by . The inverse is used to find the solution to a system of linear equation. Using determinant and adjoint, we can easily find the inverse of a square matrix.

The following results are extremely important:

1. Only a non-singular matrix can possess inverse i.e. a square matrix A possesses inverse if and only if  determinant |A|  0.Then A is said to be invertible.
2. The inverse of a matrix, where exists, is unique i.e. a non-singular matrix A cannot possess different inverse, say B and C. If A is a non-singular matrix, then 

Algorithm to find inverse of a matrix: 

Suppose a square matrix A is given whose inverse is to be obtained.

1. Find |A|. If |A| = 0, write “Inverse does not exist”. If |A|  0 write “Inverse exists” and proceed to step 2.
2. Find cofactor of all elements of A.
3. Write matrix of the cofactor of A.
4. Write adj A
5. 
6. Whether the inverse is correct verify it by  = I (Identitiy Matrix).

Suppose a 2*2 matrix A whose determinant is not equal to 0.  where a,b,c,d are number, the inverse is 

Example 1: Find the inverse of the following matrix : 

Solution :  = 16*5 – (-10)(-8) = 80 -80 = 0.

does not exist as |B| 0.

Example 2: 

Find the inverse of matrix 

Solution:  = -4 – (-3) = -1  |A|  0  and    exists.

The minor of the element are :  = -2  = -2,  = 1 = 1,  = -3 = +3,  = 2 = 2.

Cofactor matrix =   and Adj A = 

  =  = .

Exercise

1. Find the inverse of the matrix .
2. Find the inverse of matrix .
3. Find the inverse of matrix.
4. Find the inverse of matrix .
5. Find the inverse of the matrix .