= 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.
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