# 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^{−1}A^{−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^{T}A^{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.