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# Definition of inverse matrix of a square matrix

Now, we define the inverse 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 = In , 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 = In and AC = CA = In

Using these equations, we get

C = CIn = C( AB) = (CA)B = InB = B.

Hence the uniqueness follows.

Notation The inverse of a matrix A is denoted by AŌłÆ1.

Note

AAŌłÆ1 = AŌłÆ1 A = In .

Theorem 1.3

Let A be square matrix of order n. Then, AŌłÆ1 exists if and only if A is non-singular.

Proof

Suppose that AŌłÆ1 exists. Then AAŌłÆ1 = AŌłÆ1 A = In .

By the product rule for determinants, we get

det( AAŌłÆ1 ) = det( A) det( AŌłÆ1 ) = det( AŌłÆ1 ) det( A) = det(In ) = 1. So, |A| = det( A) ŌēĀ 0.

Hence A is non-singular.

Conversely, suppose that A is non-singular.

Then |A | ŌēĀ 0. By Theorem 1.1, we get Thus, we are able to find a matrix B = 1/|A| adj A such that AB = BA = In .

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 AŌłÆ1

Solution

We first find adj A. By definition, we get ### Example 1.3

Find the inverse of the matrix Solution Tags : Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix , 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants
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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants : Definition of inverse matrix of a square matrix | Definition, Theorem, Formulas, Solved Example Problems | Inverse of a Non-Singular Square Matrix