Under the assumption that the polynomial is rational and irreducible, we compute the asymptotic number of rational matrices of the given characteristic polynomial, thus solving a new case of Manin's Conjecture. The method of proof is inspired by the counting lattice point theorem of Eskin-Mozes-Shah. Here, the above rational matrices are a single orbit under the rational points of $PGL_n$, which is a lattice in the adelic points of $PGL_n$. However, we do not make use of unipotent flows and Ratner's theorems. Instead, we will prove an equidistribution theorem for an average over periodic torus orbits, relying on the measure rigidity theorem of Einsiedler-Katok-Lindenstrauss.