Let $M$ be a finite matroid with ground set $E$. Then $M$ is regular, meaning that $M$ is representable over every field, if and only if there exist $r \in \mathbb{Z}_{\ge 0}$ and an integer matrix $A \in \mathbb{Z}^{r \times E}$ whose columns are indexed by $E$ such that $A$ represents $M$ over $\mathbb{Q}$ and $A$ is totally unimodular, meaning that every square subdeterminant of $A$ belongs to $\{-1,0,1\}$.