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  "body": "\n## Definition\n\nLet $A$ be an $m \\times n$ [[matrix|0VD9ZSWWJG]] with entries in $\\Z$.\nThe matrix $A$ is *totally unimodular* when every [[minor|0BT9BYWVRY]] of $A$, that is, the determinant of every square submatrix, is $0$, $1$ or $-1$.\n\n## Remarks\n\nExample: $\\begin{pmatrix} 1 \u0026 1 \u0026 0 \\\\ 0 \u0026 1 \u0026 1 \\end{pmatrix}$ is totally unimodular.\nIts $1 \\times 1$ minors are entries in $\\{0, 1\\}$, and its $2 \\times 2$ minors are $1$, $1$ and $1$.\nNon-example: $\\begin{pmatrix} 1 \u0026 1 \\\\ -1 \u0026 1 \\end{pmatrix}$ is not totally unimodular, because its determinant is $2$.\nA totally unimodular matrix has all entries in $\\{0, 1, -1\\}$, since the entries are its $1 \\times 1$ minors.\nA square totally unimodular matrix need not be [[unimodular|R6RBQHW191]], because its determinant may be $0$.\nThe incidence matrix of a [[bipartite graph|0ZVQKTRKVQ]] and the incidence matrix of a [[directed graph|0KFYGSHHF5]] are totally unimodular.\nBy a theorem of Hoffman and Kruskal, an integer matrix $A$ is totally unimodular exactly when the polyhedron $\\set{x \\in \\R^n : Ax \\le b,\\ x \\ge 0}$ has integral vertices for every integral vector $b$.\n",
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