If \(A = \begin{bmatrix}3 & 4\\ 2 & 3\end{bmatrix}\), then \(A + A^{-1}\) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(6I\)
\(|A| = 3 \times 3 - 4 \times 2 = 1\)
For a 2 × 2 matrix, \(A^{-1} = \displaystyle\frac{1}{|A|}\,\mathrm{adj}\,A\), and the adjoint swaps the diagonal entries and changes the signs of the other two:
\(\displaystyle A^{-1} = \begin{bmatrix}3 & -4\\ -2 & 3\end{bmatrix}\)
\(\displaystyle A + A^{-1} = \begin{bmatrix}3 + 3 & 4 - 4\\ 2 - 2 & 3 + 3\end{bmatrix}\)
\(\displaystyle A + A^{-1} = \begin{bmatrix}6 & 0\\ 0 & 6\end{bmatrix} = 6I\)
Why the other options are incorrect
Option (1) O — Incorrect. The off-diagonal entries cancel, but the diagonal entries add to 6.
Option (2) \(\begin{bmatrix}6 & 8\\ 4 & 6\end{bmatrix}\) — Incorrect. This is 2A; it takes \(A^{-1}\) equal to A, without changing the signs of 4 and 2.
Option (3) \(\begin{bmatrix}0 & 8\\ 4 & 0\end{bmatrix}\) — Incorrect. This is \(A - A^{-1}\), not the sum.