This is not true in general.
$\left(\begin{array}{ccc} 4 & 0 & -16 \\ 2 & 4 & 0 \\ 0 & 2 & 4 \end{array}\right)= \left(\begin{array}{ccc} 1 & -2 & -16 \\ 0 & 1 & -2 \\ 0 & 0 & 1 \end{array}\right) +\left(\begin{array}{ccc} 3 & 2 & 0 \\ 2 & 3 & 2 \\ 0 & 2 & 3 \end{array}\right)$
A matrix with determinant $0$ is $M+S$. (Thanks to S. Sra for the correction. I multiplied by $4$ so everything's an integer now.)