If \(A=\begin{bmatrix}3 & 2\\ -1 & 1\end{bmatrix}\) and \(B=\begin{bmatrix}-1 & 0\\ 2 & 5\\ 3 & 4\end{bmatrix}\), then \((BA)^T\) is equal to:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\begin{bmatrix}-3 & 1 & 5\\ -2 & 9 & 10\end{bmatrix}\)
B is 3 × 2 and A is 2 × 2, so BA is 3 × 2. Multiplying rows of B by columns of A:
\( \displaystyle BA = \begin{bmatrix}-1 & 0\\ 2 & 5\\ 3 & 4\end{bmatrix}\begin{bmatrix}3 & 2\\ -1 & 1\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}-3 + 0 & -2 + 0\\ 6 - 5 & 4 + 5\\ 9 - 4 & 6 + 4\end{bmatrix} = \begin{bmatrix}-3 & -2\\ 1 & 9\\ 5 & 10\end{bmatrix} \)
The transpose turns rows into columns, giving a 2 × 3 matrix:
\( \displaystyle (BA)^T = \begin{bmatrix}-3 & 1 & 5\\ -2 & 9 & 10\end{bmatrix} \)
Why the other options are incorrect
Option (1) \(\begin{bmatrix}3 & 1 & 5\\ 2 & 9 & 10\end{bmatrix}\) — Incorrect. The first row of BA is (−3, −2), so the first column of the transpose is −3, −2, not 3, 2.
Option (3) \(\begin{bmatrix}-3 & -2\\ 1 & 9\\ 5 & 10\end{bmatrix}\) — Incorrect. This is BA itself; the transpose has not been taken.
Option (4) \(\begin{bmatrix}3 & 2\\ 1 & 9\\ 5 & 10\end{bmatrix}\) — Incorrect. This is BA with the signs of its first row changed, and it is not transposed.