Let the matrix be \(A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}\)

The determinant of \(A\) is calculated as follows:

\(\text{Determinant} = 1 \cdot 5 - 2 \cdot 3 = 5 - 6 = -1\)

To find the inverse of \(A\), we first determine the adjoint matrix:

\(\text{Adjoint} = \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix}\)

The inverse is given by

\(A^{-1} = \frac{1}{\text{Determinant}} \cdot \text{Adjoint} = \frac{1}{-1} \begin{bmatrix} 5 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -5 & 2 \\ 3 & -1 \end{bmatrix}\)

Next, we calculate the sum of the elements in the inverse matrix:

\(\text{Sum} = -5 + 2 + 3 + (-1) = -1\)