The inverse of the matrix […] is

The correct answer is Option (3) → \(\displaystyle\frac{1}{2}\begin{bmatrix}2 & -4\\ -1 & 3\end{bmatrix}\)

Let \(A = \begin{bmatrix}3 & 4\\ 1 & 2\end{bmatrix}\). Then

    \( |A| = 3 \times 2 - 4 \times 1 = 2 \ne 0 \)

so A is invertible. For a 2 × 2 matrix, the adjoint is found by swapping the diagonal entries and changing the signs of the other two:

    \( \displaystyle \mathrm{adj}\,A = \begin{bmatrix}2 & -4\\ -1 & 3\end{bmatrix} \)

Using \(\displaystyle A^{-1} = \frac{1}{|A|}\,\mathrm{adj}\,A\):

    \( \displaystyle A^{-1} = \frac{1}{2}\begin{bmatrix}2 & -4\\ -1 & 3\end{bmatrix} \)

Check: \(\displaystyle \begin{bmatrix}3 & 4\\ 1 & 2\end{bmatrix}\begin{bmatrix}2 & -4\\ -1 & 3\end{bmatrix} = \begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}\), and half of this is I.

Why the other options are incorrect

• Option (1) \(-\displaystyle\frac{1}{2}\begin{bmatrix}2 & -4\\ -1 & 3\end{bmatrix}\) — Incorrect. It uses \(|A| = -2\); the determinant is \(6 - 4 = 2\).

• Option (2) \(\begin{bmatrix}2 & -4\\ -1 & 3\end{bmatrix}\) — Incorrect. This is adj A; it has not been divided by \(|A| = 2\).

• Option (4) \(\displaystyle\frac{1}{2}\begin{bmatrix}3 & -4\\ -1 & 2\end{bmatrix}\) — Incorrect. The diagonal entries 3 and 2 have not been swapped.