= Integer inverse criterion for a unimodular matrix
{title2=$U\in\mathbb Z^{n\times n},\quad U^{-1}\in\mathbb Z^{n\times n}\iff\det U=\pm1$}
For a square <integer matrix> $U$ with nonzero <determinant>, \b[the <inverse matrix> has integer entries exactly when $\det U=\pm1$]. The <adjugate matrix> formula proves one direction:
$$
U^{-1}=\frac{\operatorname{adj}(U)}{\det U}.
$$
Every entry of $\operatorname{adj}(U)$ is an integer <cofactor>, so division by $1$ or $-1$ leaves integer entries. Conversely, if both $U$ and $U^{-1}$ are <integer matrices>, their <determinants> are <integers> with product one. The only possibilities are that both are one or both are minus one. This explains why a general <integer matrix> with nonzero <determinant> need not have an integer <inverse matrix>.
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