Two Classes Of Matrix Equations Over Finite Fields

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields.We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials.We also find a generating set for the vanishing ideal of these orbits.Finally,we derive a formula for the cardinality of solutions of another family of cubic matrix equations over finite fields.