Integer inverse criterion for a unimodular matrix ($U\in {\mathbb{Z}}^{n\times n},U^{-1}\in {\mathbb{Z}}^{n\times n}⟺\det U=\pm 1$)

For a square integer matrix $U$ with nonzero determinant, the inverse matrix has integer entries exactly when $\det U=\pm 1$. The adjugate matrix formula proves one direction:Every entry of $\mathrm{adj}(U)$ 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.