The inverse of the matrix \(A=\begin{bmatrix}2 & 3\\ 1 & 2\end{bmatrix}\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\begin{bmatrix}2 & -3\\ -1 & 2\end{bmatrix}\)
The inverse is \(A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,A\).
\(|A| = 2 \times 2 - 3 \times 1 = 1\)
For a 2 × 2 matrix, the adjoint is found by swapping the diagonal entries and changing the signs of the other two:
\( \mathrm{adj}\,A = \begin{bmatrix}2 & -3\\ -1 & 2\end{bmatrix} \)
Since |A| = 1,
\( A^{-1} = \begin{bmatrix}2 & -3\\ -1 & 2\end{bmatrix} \)
Check: \(AA^{-1} = \begin{bmatrix}4-3 & -6+6\\ 2-2 & -3+4\end{bmatrix} = I\).