Let $M_{n,m}:=Ma{t}_n(\mathbb{Z}/m\mathbb{Z})$ be the ring of matrices of $n\times n$ over $\mathbb{Z}/m\mathbb{Z}$ and $G_{n,m}:=G{l}_n(\mathbb{Z}/m\mathbb{Z})$ be the multiplicative group of units of $M_{n,m}$ with $n⩾2,m⩾2.$ In this paper, we obtain an exact formula for the number of representations of any element of $M_{2,m}$ as the sum of $k$ units in $M_{2,m}$. Furthermore, by using the technique of Fourier transformation, we also give a formula for the case $n\ge 3$ and $m=p$ is a prime.

Let ${\mathbb{F}}_q$ be a finite field of $q$ elements. For $n\in {\mathbb{N}}^{\ast }$ with $n\ge 2,$ let $M_n:=Ma{t}_n({\mathbb{F}}_q)$ be the ring of matrices of order $n$ over ${\mathbb{F}}_q,$ $G_{n,1}:=S{l}_n({\mathbb{F}}_q)$ be the special linear group over ${\mathbb{F}}_q.$ In this paper, by using the technique of Fourier transformation, we obtain a formula for the number of representations of any element of $M_n$ as the sum of $k$ matrices in $G_{n,1}.$ As a corollary, we give another proof of the number of the third power moment of the classic Kloosterman sum.