Question

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

(a) True

(b) False

This question was posed to me during an interview.

Origin of the question is Invertible Matrices in chapter Matrices of Mathematics – Class 12

Answer 1

The correct answer is (a) True

Explanation: The given statement is true.

(AB) (AB)^-1=I (Using the formula AA^-1=I)

Multiplying both sides by A^-1, we get

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

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

IB(AB^-1)=A^-1

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

⇒B^-1 B(AB^-1)=B^-1 A^-1

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