1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{pmatrix} 3 & 5 \\ 4 & 1 \end{pmatrix}$$. 2. **Formula for Inverse of a 2x2 Matrix:** For a matrix $$M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$, the inverse is given by: $$M^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ provided that the determinant $$ad - bc \neq 0$$. 3. **Calculate the determinant:** $$\det(A) = (3)(1) - (5)(4) = 3 - 20 = -17$$ Since $$\det(A) \neq 0$$, the inverse exists. 4. **Apply the formula:** $$A^{-1} = \frac{1}{-17} \begin{pmatrix} 1 & -5 \\ -4 & 3 \end{pmatrix} = \begin{pmatrix} -\frac{1}{17} & \frac{5}{17} \\ \frac{4}{17} & -\frac{3}{17} \end{pmatrix}$$ 5. **Final answer:** The inverse matrix is $$A^{-1} = \begin{pmatrix} -\frac{1}{17} & \frac{5}{17} \\ \frac{4}{17} & -\frac{3}{17} \end{pmatrix}$$.