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

Chapter: 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

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 A1.

Note

AA1 = A1 A = In .

 

Theorem 1.3

Let A be square matrix of order n. Then, A1 exists if and only if A is non-singular.

Proof

Suppose that A1 exists. Then AA1 = A1 A = In .

By the product rule for determinants, we get

det( AA1 ) = det( A) det( A1 ) = det( A1 ) det( A) = det(In ) = 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| In.


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 A−1

Solution

We first find adj A. By definition, we get 


 

Example 1.3

Find the inverse of the matrix 

Solution


 

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12th Mathematics : UNIT 1 : Applications of Matrices and Determinants


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