Question

P and Q are two square matrices of the same order. Which of the following statement(s) is/are correct?

• A. If P and Q are invertible, then \( [PQ]^{-1} = Q^{-1} P^{-1} \).
• B. If P and Q are invertible, then \( [QP]^{-1} = P^{-1} Q^{-1} \).
• C. If P and Q are invertible, then \( [PQ]^{-1} = P^{-1} Q^{-1} \).
• D. If P and Q are not invertible, then \( [PQ]^{-1} = Q^{-1} P^{-1} \).

Hint:

The inverse of a product of two matrices is given by \( (PQ)^{-1} = Q^{-1} P^{-1} \) and \( (QP)^{-1} = P^{-1} Q^{-1} \), but the order of multiplication must be reversed when dealing with matrix inverses.

Answer 1

The Correct Option is A, B

Let’s analyze the properties of matrix inverses for the given options: - Option (A) If P and Q are invertible, then \( [PQ]^{-1} = Q^{-1} P^{-1} \):
This is correct. The inverse of a product of two matrices is equal to the product of the inverses of the matrices, taken in reverse order. That is: \[ (PQ)^{-1} = Q^{-1} P^{-1} \] - Option (B) If P and Q are invertible, then \( [QP]^{-1} = P^{-1} Q^{-1} \):
This is correct. Similarly, the inverse of the product of matrices \( QP \) is: \[ (QP)^{-1} = P^{-1} Q^{-1} \] This follows from the same rule as in option (A). - Option (C) If P and Q are invertible, then \( [PQ]^{-1} = P^{-1} Q^{-1} \):
This is incorrect. This would be true only if we were multiplying \( P^{-1} \) and \( Q^{-1} \) in the reverse order, as shown in option (A). The correct order is \( Q^{-1} P^{-1} \), not \( P^{-1} Q^{-1} \). - Option (D) If P and Q are not invertible, then \( [PQ]^{-1} = Q^{-1} P^{-1} \):
This is incorrect. If \( P \) or \( Q \) is not invertible, then \( PQ \) will also not be invertible, and the inverse does not exist. Thus, the correct answers are Option (A) and Option (B).