To find the inverse of the matrix $A = \begin{bmatrix} -2 & 3 \\ -2 & 4 \end{bmatrix}$, we follow these steps:

1. Find the determinant of the matrix: $\text{det}(A) = (-2)(4) - (-2)(3) = -8 + 6 = -2$

2. Compute the adjugate matrix: Swap the diagonal elements and negate the off-diagonal elements: $\text{Adj}(A) = \begin{bmatrix} 4 & -3 \\ 2 & -2 \end{bmatrix}$

3. Divide the adjugate matrix by the determinant: $A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{Adj}(A)$ Substituting $\text{det}(A) = -2$:

   = \begin{bmatrix} -2 & \frac{3}{2} \\ -1 & 1 \end{bmatrix}$$

The correct inverse is: $\boxed{\text{Option A: } \begin{bmatrix} -2 & \frac{3}{2} \\ -1 & 1 \end{bmatrix}}$

Would you like more details on how the adjugate or determinant was calculated?

Related Questions:

1. What is the determinant, and why is it important for inverses?
2. Can all square matrices be inverted?
3. What happens if the determinant is zero?
4. How is the adjugate matrix derived systematically?
5. How can this process be extended to larger matrices?

Tip:

For 2x2 matrices, the inverse formula is quick: swap diagonals, negate off-diagonals, and divide by the determinant.