Question

If A is an invertible matrix of order 2, then det ( A −1 ) is equal to A. det ( A ) B. C. 1 D. 0

Answer 1

Let A=[ a b c d ]

The determinant of A is,

| A |=ad−bc

The adjoint of A is,

adjA=[ d −c −b a ]

It is given that A is an invertible matrix, so, A −1 exists and A −1 = 1 | A | adjA.

Substitute the values of | A | and adjA in above formula.

A −1 = 1 | A | [ d −c −b a ] A −1 =[ d | A | −c | A | −b | A | a | A | ] | A −1 |=| d | A | −c | A | −b | A | a | A | | | A −1 |= 1 | A | 2 | d −c −b a |

Simplify further,

| A −1 |= 1 | A | 2 ( ad−bc ) | A −1 |= 1 | A | 2 | A | | A −1 |= 1 | A | det( A −1 )= 1 det( A )

Hence, option (B) is the correct.