Invertible Matrices

A matrix is an array of numbers arranged in the form of rows and columns. The number of rows and columns of a matrix are known as its dimensions, which is given by m x n where m and n represent the number of rows and columns respectively. The basic mathematical operations like addition, subtraction, multiplication and division can be done on matrices. In this article, we will discuss the inverse of a matrix or the invertible vertices.

What is an Invertible Matrices?

A matrix A of dimension n x n is called invertible if and only if there exists another matrix B of the same dimension, such that AB = BA = I, where I is the identity matrix of the same order. Matrix B is known as the inverse of matrix A. Inverse of matrix A is symbolically represented by A-1.

For example, matrices A and B are given below:

matrices

matrices

Now we multiply A with B and obtain an identity matrix:

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Similarly, on multiplying B with A, we obtain the same identity matrix:

matrices

It can be concluded here that AB = BA = I. Hence A-1 = B, and B is known as the inverse of A. Similarly, A can also be called an inverse of B, or B-1 = A.

Theorems

  • Theorem 1: If there exists an inverse of a square matrix, it is always unique.

Proof: Let us take A to be a square matrix of order n x n. Let us assume matrices B and C to be inverses of matrix A.

Now AB = BA = I since B is the inverse of matrix A.

Similarly, AC = CA = I.

But, B = BI = B (AC) = (BA) C = IC = C

This proves B = C, or B and C are the same matrices.

  • Theorem 2: If A and B are matrices of the same order and are invertible, then (AB)-1 = B-1 A-1.

Proof:

(AB)(AB)-1 = I                                     (From definition of inverse of a matrix)

A-1 (AB)(AB)-1 = A-1 I                         (Multiplying A-1 on both sides)

(A-1 A) B (AB)-1 = A-1                                   (A-1 I = A-1 )

I B (AB)-1 = A-1

B (AB)-1 = A-1

B-1 B (AB)-1 = B-1 A-1

I (AB)-1 = B-1 A-1

(AB)-1 = B-1 A-1

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Practise This Question

If A = 121112211, then det (Adj (Adj A)) is