Definition of inverse matrix of a square matrix

Definition of inverse matrix of a square matrix

Now, we define the inverse of a square matrix.

Definition 1.2

Let A be a square matrix of order n. If there exists a square matrix B of order n such that AB = BA = I_n , then the matrix B is called an inverse of A.

Theorem 1.2

If a square matrix has an inverse, then it is unique.

Proof

Let A be a square matrix order n such that an inverse of A exists. If possible, let there be two inverses B and C of A. Then, by definition, we have AB = BA = I_n and AC = CA = I_n

Using these equations, we get

C = CI_n = C( AB) = (CA)B = I_nB = B.

Hence the uniqueness follows.

Notation The inverse of a matrix A is denoted by A^−1.

Note

AA^−1 = A^−1 A = I_n .

Theorem 1.3

Let A be square matrix of order n. Then, A^−1 exists if and only if A is non-singular.

Proof

Suppose that A^−1 exists. Then AA^−1 = A^−1 A = I_n .

By the product rule for determinants, we get

det( AA^−1 ) = det( A) det( A^−1 ) = det( A^−1 ) det( A) = det(I_n ) = 1. So, |A| = det( A) ≠ 0.

 Hence A is non-singular.

Conversely, suppose that A is non-singular.

Then |A | ≠ 0. By Theorem 1.1, we get

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

Thus, we are able to find a matrix B = 1/|A| adj A such that AB = BA = I_n .

Hence, the inverse of A exists and it is given by 

Remark

The determinant of a singular matrix is 0 and so a singular matrix has no inverse.

Example 1.2

If A =  is non-singular, find A^−1

Solution

We first find adj A. By definition, we get 

Example 1.3

Find the inverse of the matrix 

Solution