If \(A=\begin{bmatrix}5 & 1\\ -2 & 0\end{bmatrix}\) and \(B^T=\begin{bmatrix}1 & 10\\ -2 & -1\end{bmatrix}\), then the matrix AB is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\begin{bmatrix}15 & -11\\ -2 & 4\end{bmatrix}\)
Since \(\left(B^T\right)^T = B\), B is found by turning the rows of \(B^T\) into columns:
\( \displaystyle B = \begin{bmatrix}1 & -2\\ 10 & -1\end{bmatrix} \)
Multiplying rows of A by columns of B:
\( \displaystyle AB = \begin{bmatrix}5 & 1\\ -2 & 0\end{bmatrix}\begin{bmatrix}1 & -2\\ 10 & -1\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}5 + 10 & -10 - 1\\ -2 + 0 & 4 + 0\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}15 & -11\\ -2 & 4\end{bmatrix} \)