@article{Luo2025, 
author = {Yifan Luo and Qingzhong Ji},
title = {On the sum of matrices of special linear group over finite field},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {2},
pages = {3642-3651},
keywords = {ring of matrices, finite field, sum of matrices, Fourier transformation, Kloosterman sum},
url = {https://www.sciopen.com/article/10.3934/math.2025168},
doi = {10.3934/math.2025168},
abstract = {Let              F        q   be a finite field of    q elements. For    n  ∈            N        ∗   with    n  ≥  2  , let        M          n        :=  M  a      t    n    (            F        q    ) be the ring of matrices of order    n over              F        q    ,         G          n      ,      1        :=  S      l    n    (            F        q    ) be the special linear group over              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.}
}