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Inverse of a Matrix

The multiplicative inverse of a square matrix is called its inverse matrix. If a matrix A has an inverse, then A is said to be nonsingular or invertible. A singular matrix does not have an inverse. To find the inverse of a square matrix A , you need to find a matrix A − 1 such that the product of A and A − 1 is the identity matrix.

In other words, for every square matrix A which is nonsingular there exist an inverse matrix, with the property that, A A − 1 = A − 1 A = I , where I is the identity matrix of the appropriate size.

You can use either of the following method to find the inverse of a square matrix.

Method 1:

Let A be an n × n matrix.

1. Write the doubly augmented matrix [ A   |   I n ] .

2. Apply elementary row operations to write the matrix in reduced row-echelon form.

3. Decide whether the matrix A is invertible (nonsingular).

4. If A can be reduced to the identity matrix I n , then A − 1 is the matrix on the right of the transformed augmented matrix.

5. If A cannot be reduced to the identity matrix, then A is singular.

Method 2:

You may use the following formula when finding the inverse of n × n matrix.

If A is non-singular matrix, there exists an inverse which is given by A − 1 = 1 |   A   | ( adj  A ) , where |   A   | is the determinant of the matrix.

Example :

Find A − 1 , if it exists. If A − 1 does not exist, write singular.

A = [ 1 2 1 1 ]

Step 1:

Write the doubly augmented matrix [ A   |   I n ] .

[ A   |   I ] = [ 1 2 1 0 1 1 0 1 ]

Step 2:

Apply elementary row operations to write the matrix in reduced row-echelon form.

[ 1 2 1 0 0 1 1 − 1 ] R 2 = R 1 − R 2 [ 1 0 − 1 2 0 1 1 − 1 ] R 1 = − 2 R 2 + R 1 [ 1 0 − 1 2 0 1 1 − 1 ] = [ I   |   A − 1 ]

The system has a solution.

Therefore, A is invertible and A − 1 = [ − 1 2 1 − 1 ]