A quadratic form tr(AX2) and its application

We give the complete classification of quadratic forms tr(AX2) obtained by taking the trace of n × n matrices AX2 for n = 2,3, where A ε Mn(Fq), X is a square matrix of n2 variables, and Fq is a finite field of order q and of characteristic different from 2. As an application of the classification we can compute the number of solutions of a matrix equation X21 + X22 + … + X2m = B for a 2 × 2 matrix B over Fq.