Distribution of determinant of sum of matrices

Abstract

Let Fq be an arbitrary finite field of order q. In this article, we study det S for certain types of subsets S in the ring M2(Fq) of 2 × 2 matrices with entries in Fq. For i ∈ Fq, let Di be the subset of M2(Fq) defined by Di := {x ∈ M2(Fq) : det(x) = i}. Then our results can be stated as follows. First of all, we show that when E and F are subsets of Di and Dj for some i, j ∈ F * q , respectively, we have det(E + F ) = Fq,