Unimodular Matrices and Loops

A unimodular matrix is a square integer matrix with determinant $\pm 1$. With some basic linear algebra lemmas, we can show that a square integer matrix is unimodular if and only if its inverse exists and the inverse matrix is also an integer matrix.

Let $A$ be an unimodular matrix. As the determinant of an integer matrix must also be an integer, $\operatorname{adj}(A)$ is also an integer matrix. It follows that $A^{-1} = \frac{1}{\det{A}} \operatorname{adj}(A)$ is an integer matrix.

To show the converse, let $A$ be an integer square matrix such that $A^{-1}$ is an integer matrix as well. Since $\det(A) \det(A^{-1}) = \det(A A^{-1}) = \det(I) = 1$ and $\det(A), \det(A^{-1}) \in \mathbb Z$, we can conclude that $A$ and $A^{-1}$ are unimodular.