If \(P=\begin{bmatrix}-1\\ 2\\ 1\end{bmatrix}\) and \(Q=\begin{bmatrix}2 & -4 & 1\end{bmatrix}\) are two matrices, then \((PQ)'\) will be:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\begin{bmatrix}-2 & 4 & 2\\ 4 & -8 & -4\\ -1 & 2 & 1\end{bmatrix}\)
P is 3 × 1 and Q is 1 × 3, so PQ is 3 × 3. Each entry is (row entry of P) × (column entry of Q):
\( PQ = \begin{bmatrix}-1\\ 2\\ 1\end{bmatrix}\begin{bmatrix}2 & -4 & 1\end{bmatrix} \)
\( = \begin{bmatrix}-2 & 4 & -1\\ 4 & -8 & 2\\ 2 & -4 & 1\end{bmatrix} \)
The transpose turns rows into columns:
\( (PQ)' = \begin{bmatrix}-2 & 4 & 2\\ 4 & -8 & -4\\ -1 & 2 & 1\end{bmatrix} \)
Shortcut: By the reversal law, (PQ)′ = Q′P′, a column (2, −4, 1) times the row (−1, 2, 1). Its first row is 2 × (−1, 2, 1) = (−2, 4, 2), and only Option (2) has this first row.