Question

Invertible Matrices

Answer 1

Definition : If A is a square matrix of order m and if there exists another square matrix B of the same order m, such that AB = BA = I, then B is called the inverse matrix of A and it is denoted by A^-1. In that case A is said to be invertible.

Thus, B is the inverse of A,in other words B = A^-1 and A is inverse of B, i.e., A = B^-1

Note:

• A rectangular matrix does not possess inverse matrix, since for products BA and AB to be defined and to be equal, it is necessary that matrices A and B should be square matrices of the same order.
• If B is the inverse of A, then A is also inverse of B.

Theorem 3.

(Uniqueness of inverse) Inverse of a square matrix, if it exists, is unique.

Proof :

Let A = [a_ij] be a square matrix of order m. If possible, let B and C be two inverses of A. We shall show that B = C.

Since, B is the inverse of A

AB = BA = I ...(1)

Since, C is also the inverse of A

AC = CA = I ...(2)

Thus B = BI = B(AC)

= (BA)C = IC = C

Theorem 4.

If A and B are invertible matrices of the same'order, then (AB)^-1 = B^-1 A^-1.

Proof :

From the definition of inverse of a matrix, we have (AB) (AB)^-1 = 1 ‘

or A^-1 (AB) (AB)^-1 = A^-1I (Pre multiplying both sides by A^-1)

or (A^-1A) B (AB)^-1 = A^-1(sine A^-1I = A^-1)

or IB (AB)^-1 = A^-1

or B(AB)^-1 = A^-1

or B^-1B (AB)^-1 = B^-1 A^-1

or I (AB)^-1 = B^-1 A^-1

Hence (AB)^-1 = B^-1 A^-1