$\left(\begin{array}{ccc} 1 & 0 & -4 \\ 1/2 & 1 & 0 \\ 0 & 1/2 & 1 \end{array}\right)= \left(\begin{array}{ccc} 1 & -1/2 & -4 \\ 0 & 1 & -1/2 \\ 0 & 0 & 1 \end{array}\right) +\left(\begin{array}{ccc} 1 & 1/2 & 0 \\ 1/2 & 1 & 1/2 \\ 0 & 1/2 & 1 \end{array}\right)$
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 $1,1,1$, $3/4, 1, 3/4$, $1/2$.