If \(A = \begin{bmatrix}5 & -2\\ 3 & 4\end{bmatrix}\), then \(\mathrm{adj}\,A\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}4 & 2\\ -3 & 5\end{bmatrix}\)
For a \(2 \times 2\) matrix, the adjoint is found by swapping the diagonal elements and changing the signs of the other two:
\( \displaystyle \mathrm{adj}\begin{bmatrix}a & b\\ c & d\end{bmatrix} = \begin{bmatrix}d & -b\\ -c & a\end{bmatrix} \)
With a = 5, b = −2, c = 3, d = 4:
\( \displaystyle \mathrm{adj}\,A = \begin{bmatrix}4 & 2\\ -3 & 5\end{bmatrix} \)
Why the other options are incorrect
Option (1) \(\begin{bmatrix}4 & -2\\ 3 & 5\end{bmatrix}\) — Incorrect. The diagonal is swapped, but the signs of −2 and 3 are not changed.
Option (2) \(\begin{bmatrix}-4 & 2\\ 3 & -5\end{bmatrix}\) — Incorrect. The signs of the diagonal elements are changed; only the off-diagonal signs change.
Option (4) \(\begin{bmatrix}5 & 3\\ -2 & 4\end{bmatrix}\) — Incorrect. This is \(A^{T}\), the transpose of A.