Question

Given $P$ is an $m\times n$ matrix, $Q$ is an $n\times l$ matrix, and $R$ and $S$ are $n\times n$ matrices. Consider:
Relationship 1: \quad $(PQ)^T = Q^T P^T$
Relationship 2: \quad $(RS)^{-1} = S^{-1} R^{-1}$
Which one of the following is correct?

• A. Relationship 1 is false; Relationship 2 is false
• B. Relationship 1 is true; Relationship 2 is false
• C. Relationship 1 is true; Relationship 2 is true
• D. Relationship 1 is false; Relationship 2 is true

Hint:

Both transpose and inverse reverse multiplication order: $(AB)^T = B^T A^T$ and $(AB)^{-1} = B^{-1} A^{-1}$.

Answer 1

The Correct Option is C

Relationship 1: Transpose of a product reverses the order: \[ (PQ)^T = Q^T P^T. \] This is a standard matrix identity. Thus, Relationship 1 is true. Relationship 2: Inverse of a product also reverses the order: \[ (RS)^{-1} = S^{-1} R^{-1}, \] provided $R$ and $S$ are invertible. This is also a well-known identity. Thus, Relationship 2 is true. Therefore, both relationships are correct. Final Answer: Relationship 1 true, Relationship 2 true