 Signed column condition for total unimodularity

A matrix with entries in $\{0,1,-1\}$ is a totally unimodular matrix if every column has at most two nonzero entries, with opposite signs when both occur. To prove this, induct on the size of a square submatrix. A zero column gives determinant zero, a one-entry column reduces by cofactor expansion, and if every column has two entries then all column sums vanish, giving dependent rows. The total unimodularity of directed incidence matrices is a central instance.