If the matrix has integer elements, then the geometric meaning is that a unimodular transformation maps the integer lattice onto itself.

If the matrix has integer elements, then the geometric meaning is that a unimodular transformation maps the integer lattice onto itself:

Consider a basis $B$ of an $m$-dimensional lattice $L(B)=\{Bx:x\in\mathbb{Z}^m$}, and another basis $C$, then $L(B)=L(C)$ if and only if there exists a unimodular matrix $M$ (an $m\times m$ matrix with integer entries and determinant $\pm 1$) such that $B=CM$.