On the sum of matrices of special linear group over finite field

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.