Order of the group associated to a quotient of a lattice

Question

Let $A=[A_{ij}]$ be a $n\times n$ symmetric positive definite integer-valued matrix. Define elements of $\mathbb{Z}^n$ $v_i=[A_{i1},A_{i2},...,A_{in}]$ where I am treating them as row vectors. Quotient $\mathbb{Z}^n$ by imposing $v_i\equiv 0$. We obtain a quotient group which is a finite group $G$ because there are $n$ linearly independent relations $v_i\equiv 0$. Why is the order of $G$ equal to the determinant of $A$?

Answer 1

Show that the order of $G$ transforms in the expected way when we perform elementary row and column operations, and reduce to the case where $A$ is diagonal.