I have a matrix $M$ such that every entry in $M$ is $1,-1$ or $0$. My matrix also has the following property. Let $v_i$ and $v_j$ be two columns of $M$, and let $S$ be the set of row indices $s$ such that both $v_i$ and $v_j$ have a nonzero entry on the $s$th row. Then, noting $v_i|_S$ for the restriction of $v_i$ to the rows whose indices are in $S$, we have: $v_i|_S=\pm v_j|_S$.
My question is: is this property sufficient to prove total unimodularity of $M$? I believe the property implies that, if $N$ is some minor of $M$ whose entries are all nonzero, then the determinant of $N$ is $0$ (since its rows are all equal up to multiplication by $\pm 1$). Is this property itself sufficient for unimodularity?
Please let me know your thoughts! :)