Index of an integer sublattice

Statement

If $A$ is a nonsingular $n\times n$ integer matrix, then

\bigl|\mathbb Z^n/A\mathbb Z^n\bigr|=|\det A|.

The left side is the index of the sublattice $A\mathbb Z^n$ .

Proof by fundamental domains

The entries of $A^{-1}$ are rational. Choose a positive integer $N$ clearing their denominators. Then $N\mathbb Z^n\subset A\mathbb Z^n$ , proving that the index $k$ is finite. Take coset representatives $r_1,\ldots,r_k\in\mathbb Z^n$ . The union of $r_j+[0,1)^n$ is a fundamental domain for $A\mathbb Z^n$ of volume $k$ . The parallelepiped $A[0,1)^n$ is another, of volume $|\det A|$ . Their volumes agree: partition one domain by its intersections with lattice translates of the other, translate the pieces back, and use countable additivity. This gives the formula.