Question

Consider two matrices \[ P = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, Q = \begin{bmatrix} 5 & 4 \\ 0 & 2 \end{bmatrix}. \] If \(R = (PQ)^T\), then \(\det R\) is ............ (in integer).

Hint:

- Always remember: \(\det(A^T) = \det(A)\). - Instead of computing transpose separately, you can compute \(\det(PQ)\) directly. - Here: \(\det(PQ) = \det(P) . \det(Q)\). \(\det(P) = (2 . 4 - 3 . 1) = 5\). \(\det(Q) = (5 . 2 - 4 . 0) = 10\). So, \(\det(PQ) = 5 . 10 = 50\).

Answer 1

Correct Answer: 50

Step 1: Multiply matrices \(P\) and \(Q\). \[ PQ = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \begin{bmatrix} 5 & 4 \\ 0 & 2 \end{bmatrix} \] Perform multiplication: \[ PQ = \begin{bmatrix} (2 . 5 + 3 . 0) & (2 . 4 + 3 . 2)
(1 . 5 + 4 . 0) & (1 . 4 + 4 . 2) \end{bmatrix} = \begin{bmatrix} 10 & 14 \\ 5 & 12 \end{bmatrix} \]
Step 2: Define \(R\).
\[ R = (PQ)^T = \begin{bmatrix} 10 & 14 \\ 5 & 12 \end{bmatrix}^T = \begin{bmatrix} 10 & 5 \\ 14 & 12 \end{bmatrix} \]
Step 3: Compute determinant of \(R\).
\[ \det(R) = (10 . 12) - (5 . 14) \] \[ = 120 - 70 = 50 \] Wait, let’s double-check carefully : Earlier, I thought it was 14 — but recalc gives: \[ \det(R) = 120 - 70 = 50 \] So correct value is **50**, not 14. Final Answer: \[ \boxed{50} \]