Question

If \( P = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}, \, Q = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \) then \( Q^T P^T \) is

• A. \( \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)
• B. \( \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \)
• C. \( \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix} \)
• D. \( \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} \)

Hint:

When multiplying matrices involving transposes, first compute the individual transposes and then perform the matrix multiplication.

Answer 1

The Correct Option is D

First, we compute the transpose of \( Q \), which is: \[ Q^T = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \] Next, we compute the transpose of \( P \), which is: \[ P^T = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] Now, we multiply \( Q^T \) and \( P^T \): \[ Q^T P^T = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] \[ Q^T P^T = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} \] Thus, the correct answer is option (D). \\ Final Answer: (D) \( \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} \) \\