  In this note, we give explicit expressions of Gauss sums for general (resp.
special) linear groups over finite fields, which involves Gauss sums (resp.
Kloosterman sums). The key ingredient is averaging such sums over Borel
subgroups. As applications, we count the number of invertible matrices of
zero-trace over finite fields and we also improve two bounds by Ferguson,
Hoffman, Luca, Ostafe and Shparlinski in [ Some additive combinatorics problems
in matrix rings, Rev. Mat. Complut. (23) 2010, 501--513 ].
