1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$$. 2. **Formula for Inverse of a 2x2 Matrix:** For a matrix $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, the inverse $$A^{-1}$$ exists if and only if the determinant $$\det(A) = ad - bc \neq 0$$. The inverse is given by: $$ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} $$ 3. **Calculate the determinant:** $$ \det(A) = (2)(3) - (4)(1) = 6 - 4 = 2 $$ Since $$\det(A) = 2 \neq 0$$, the inverse exists. 4. **Apply the formula:** $$ A^{-1} = \frac{1}{2} \begin{bmatrix} 3 & -4 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} \frac{3}{2} & -2 \\ -\frac{1}{2} & 1 \end{bmatrix} $$ 5. **Final answer:** $$ A^{-1} = \begin{bmatrix} 1.5 & -2 \\ -0.5 & 1 \end{bmatrix} $$ This matrix is the inverse of $$A$$. The inverse matrix can be used to solve systems of linear equations or to find transformations that reverse the effect of $$A$$.