The inverse of the matrix \(\begin{bmatrix}3 & 5\\ 1 & 2\end{bmatrix}\) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\begin{bmatrix}2 & -5\\ -1 & 3\end{bmatrix}\)
Let \(A = \begin{bmatrix}3 & 5\\ 1 & 2\end{bmatrix}\). Then
\(|A| = 3 \cdot 2 - 5 \cdot 1 = 1\)
Since \(|A| \ne 0\), A is invertible and \( \displaystyle A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,A \).
\( \displaystyle \mathrm{adj}\,A = \begin{bmatrix}2 & -5\\ -1 & 3\end{bmatrix} \)
With \(|A| = 1\), the inverse equals the adjoint:
\( \displaystyle A^{-1} = \begin{bmatrix}2 & -5\\ -1 & 3\end{bmatrix} \)
Check: \(AA^{-1} = \begin{bmatrix}6 - 5 & -15 + 15\\ 2 - 2 & -5 + 6\end{bmatrix} = I\).