Inverse of a Matrix

Let A be a square matrix of order n. Then a matrix B, if it exists, such that AB = BA = I_n is called inverse of the matrix A. In this case, A is an invertible matrix.

If a matrix A possesses an inverse, then it must be unique.

If A is a square matrix of order n, then 

A(adj A) = (adj A)A = |A| I_n

Assume that A is non-singular, then |A| ≠ 0.

If A is a non-singular matrix, there exists an inverse which is given by

Properties

1. Reversal Law for Inverses

If A, B are any two non-singular matrices of the same order, then AB is also non-singular and

(AB)⁻¹ = B⁻¹A⁻¹

The inverse of a product is the product of the inverses taken in the reverse order.

2. Reversal Law for Transposes

If A and Bare matrices conformable to multiplication, then

(AB)^T = B^TA^T

The transpose of the product is the product of the transposes taken in the reverse order.

3. For any non-singular matrix A,

(A^T)⁻¹ = (A⁻¹)^T

Computation of Inverse

For square matrix, find |A|.

If |A| ≠ 0, then A is a non-singular matrix. Hence, it is invertible.