On the sumsets of units in a ring of matrices over $\mathbb{Z}/m\mathbb{Z}$

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.