= Alternating-cycle perturbation of a doubly stochastic matrix
For a <doubly stochastic matrix> with a fractional entry, form the row-column <bipartite graph> of entries strictly between zero and one. Every incident vertex has degree at least two, so there is an even cycle. A matrix supported on that cycle with alternating entries $+1,-1$ has zero row and column sums. A sufficiently small positive multiple can be both added and subtracted while preserving entry bounds. The original matrix is then the midpoint of two distinct doubly stochastic matrices, so it is not an <extreme point>.
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