Inverse of a Matrix
The multiplicative inverse of a square matrix is called its inverse matrix. If a matrix has an inverse, then is said to be nonsingular or invertible. A singular matrix does not have an inverse. To find the inverse of a square matrix , you need to find a matrix such that the product of and is the identity matrix.
In other words, for every square matrix which is nonsingular there exist an inverse matrix, with the property that, , where is the identity matrix of the appropriate size.
You can use either of the following method to find the inverse of a square matrix.
Let be an matrix.
1. Write the doubly augmented matrix .
2. Apply elementary row operations to write the matrix in reduced row-echelon form.
3. Decide whether the matrix is invertible (nonsingular).
4. If can be reduced to the identity matrix , then is the matrix on the right of the transformed augmented matrix.
5. If cannot be reduced to the identity matrix, then is singular.
You may use the following formula when finding the inverse of matrix.
If is non-singular matrix, there exists an inverse which is given by , where is the determinant of the matrix.
Find , if it exists. If does not exist, write singular.
Write the doubly augmented matrix .
Apply elementary row operations to write the matrix in reduced row-echelon form.
The system has a solution.
Therefore, is invertible and