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:

- 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.
- 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.

- Find |A|. If |A| = 0, write “Inverse does not exist”. If |A| 0 write “Inverse exists” and proceed to step 2.
- Find cofactor of all elements of A.
- Write matrix of the cofactor of A.
- Write adj A
- 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

- Find the inverse of the matrix .
- Find the inverse of matrix .
- Find the inverse of matrix.
- Find the inverse of matrix .
- Find the inverse of the matrix .

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