If \(A=\begin{bmatrix}3 & -5\\ 4 & -7\end{bmatrix}\), then \(A^{-1}\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\begin{bmatrix}7 & -5\\ 4 & -3\end{bmatrix}\)
For a 2 × 2 matrix, \(A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,A\), where the adjoint swaps the diagonal entries and changes the sign of the other two.
\(|A| = 3(-7)-(-5)(4) = -21+20 = -1\)
\(\mathrm{adj}\,A = \begin{bmatrix}-7 & 5\\ -4 & 3\end{bmatrix}\)
\(A^{-1} = \displaystyle\frac{1}{-1}\begin{bmatrix}-7 & 5\\ -4 & 3\end{bmatrix} = \begin{bmatrix}7 & -5\\ 4 & -3\end{bmatrix}\)
Why the other options are incorrect
Option (1) \(\begin{bmatrix}-7 & 5\\ -4 & 3\end{bmatrix}\) — Incorrect. This is adj A; it has not been divided by \(|A| = -1\).
Option (2) \(\begin{bmatrix}7 & 5\\ 4 & -3\end{bmatrix}\) — Incorrect. The (1, 2) entry should be −5.
Option (3) \(\begin{bmatrix}-7 & -5\\ -4 & 3\end{bmatrix}\) — Incorrect. Its (1, 1) entry is −7, but \(A^{-1}\) has 7 there.