Number of solutions of equations of Weil type on finite symmetric matrices

Let M2(Fq) be the ring of 2-by-2 matrices with coefficients in a finite field Fq and let Λ2(Fq) be the subset of M2(Fq) of symmetric matrices. In [4], M. Kuroda determined the number of solutions (Xi) for the equation , (B,Xi∈M2(Fq)). In this paper, we consider the problem for B,Xi∈Λ2(Fq) and give a complete formula for the number of solutions of the equation. In the process, we also give an Fq-form of Λn(Fq) for any n.