# 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, AA-1 = A-1A = 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 matrix.

1. Write the doubly augmented matrix [ A | In].
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 In, 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 matrix.

If A is non-singular matrix, there exists an inverse which is given by , where is the determinant of the matrix.

Example :

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

Step 1:

Write the doubly augmented matrix [ A | In].

Step 2:

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

The system has a solution.

Therefore, A is invertible and