If \(A = \begin{bmatrix}2 & 1\\ 3 & 2\end{bmatrix}\) and \(B = \begin{bmatrix}1 & -1\\ 2 & 3\end{bmatrix}\), then \(\mathrm{adj}(AB)\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\begin{bmatrix}3 & -1\\ -7 & 4\end{bmatrix}\)
The product is
\(\displaystyle AB = \begin{bmatrix}2 + 2 & -2 + 3\\ 3 + 4 & -3 + 6\end{bmatrix} = \begin{bmatrix}4 & 1\\ 7 & 3\end{bmatrix}\)
For a 2 × 2 matrix the adjoint is found by swapping the diagonal entries and changing the signs of the other two:
\(\displaystyle \mathrm{adj}(AB) = \begin{bmatrix}3 & -1\\ -7 & 4\end{bmatrix}\)
Check by the rule \(\mathrm{adj}(AB) = (\mathrm{adj}\,B)(\mathrm{adj}\,A)\), with the order reversed:
\(\displaystyle \begin{bmatrix}3 & 1\\ -2 & 1\end{bmatrix}\begin{bmatrix}2 & -1\\ -3 & 2\end{bmatrix} = \begin{bmatrix}3 & -1\\ -7 & 4\end{bmatrix}\)
Why the other options are incorrect
Option (2) \(\begin{bmatrix}8 & 1\\ -13 & -1\end{bmatrix}\) — Incorrect. This is \((\mathrm{adj}\,A)(\mathrm{adj}\,B)\), the product in the wrong order.
Option (3) \(\begin{bmatrix}4 & 1\\ 7 & 3\end{bmatrix}\) — Incorrect. This is AB itself, not its adjoint.
Option (4) \(\begin{bmatrix}3 & 1\\ 7 & 4\end{bmatrix}\) — Incorrect. The diagonal is swapped but the signs of the other two entries are not changed.