Let $m$ and $k$ be any positive integers, let $mathbb{Z}_k$ the ring of integers of modulo $k$, let $G_m(mathbb{Z}_k)$ the group of all $m$ by $m$ nonsingular matrices over $mathbb{Z}_k$ and let ${phi}_m(k)$ the order of $G_m(mathbb{Z}_k)$. In this paper, ${phi}_m(k)$ can be computed by the following investigation: First, for any relatively prime positive integers $s$ and $t$, $G_m(mathbb{Z}_{st})$ is isomorphic to $G_m(mathbb{Z}_s){ imes}G_m(mathbb{Z}_t)$. Secondly, for any positive integer $n$ and any prime $p$, ${phi}_m(p^n)=p^{m^2}{cdot}{phi}_m(p^{n-1})=p{^{2m}}^2{cdot}{phi}_m(p^{n-2})={cdots}=p^{{(n-1)m}^2}{cdot}{phi}_m(p)$, and so ${phi}_m(k)={phi}_m(p_1^n1){cdot}{phi}_m(p_2^{n2}){cdots}{phi}_m(p_s^{ns})$ for the prime factorization of $k$, $k=p_1^{n1}{cdot}p_2^{n2}{cdots}p_s^{ns}$.