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 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.
Example 1: Find the inverse of the following matrix :
Solution :
Example 2:
Find the inverse of matrix
Solution:
The minor of the element are :
Cofactor matrix =
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
.