Definition. Let A be a square matrix of order n. Then a square matrix B of order n, if it exists, is called an inverse of A if AB = BA = In.

A matrix A having an inverse is called an invertible matrix. It may easily be seen that if a matrix A is invertible, its inverse is unique. The inverse of a invertible matrix A is denoted by A^-1.

Does every square matrix possess an inverse  To answer this let us consider the matrix



If B is any square matrix of order 2, we find that

AB = BA = 0.

We thus see that there cannot be any matrix B for which AB and BA both are equal to I₂. Therefore A is not invertible. Hence, we con-clued that a square matrix may fail to have an inverse.

However, if A is square matrix such that |A| ≠ 0, then A is invertible and

A^-1 = 1   adj A.
        |A|

For, we know that

A (adj A) = (adj A) A = |A| In



Thus A is invertible and A^-1  = 1 adj A
                                          |A|

Singular and Non-singular Matrices. A square matrix A is said to be singular if |A| = 0, and it is called nonsingular if |A| ≠  0. For example, if



then |A| = 3 (-3+2) - 1 (2+1) +2 (4+3) = 8.
Since |A| ≠  0, A is non-singular. Further, if



then |B| = 2 (-8+8) + 0 + 0 =0
Since |B| = 0, B is singular

Properties of the Inverse of a Matrix

1.    A square matrix is invertible if and only if it is non-singular.

2.    The inverse of the inverse is the original matrix itself, i.e. (A^-1)^-1 =A

3.    The inverse of the transpose of a matrix is the transpose of its inverse, i.e. (A')^-1 = (A^-1)'.

4.    (Reversal law) if A and B are two invertible matrices of the some order, then AB is also invertible and moreover.

(AB)^-1 = B^-1 A^-1 .

For more help in Inverse of A Square Matrix click the button below to submit your homework assignment