$\left(\begin{array}{ccc} 1 & 0 & -4 \\ 1/2 & 1 & 0 \\ 0 & 1/2 & 1 \end{array}\right)= \left(\begin{array}{ccc} 1 & -1/2 & -4 \\ 0 & 1 & -1/2 \\ 0 & 0 & 1 \end{array}\right) +\left(\begin{array}{ccc} 1 & 1/2 & 0 \\ 1/2 & 1 & 1/2 \\ 0 & 1/2 & 1 \end{array}\right)$

First matrix has determinant $0$. Second matrix is an $M$-matrix because it's unipotent, so principal minors are unipotent. Third matrix is an $S$-matrix - the principal minors are $1,1,1$, $3/4, 1, 3/4$, $1/2$.

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.)