1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$. 2. **Formula for Inverse of a 2x2 Matrix:** For a matrix $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, the inverse is given by: $$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$$ provided that $$ad - bc \neq 0$$. 3. **Calculate the determinant:** $$ad - bc = (1)(4) - (2)(3) = 4 - 6 = -2$$ Since the determinant is not zero, the inverse exists. 4. **Apply the formula:** $$A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ \frac{3}{2} & -\frac{1}{2} \end{bmatrix}$$ 5. **Final answer:** $$\boxed{A^{-1} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix}}$$