  Generalizing the problem of counting rational points on curves and surfaces
over finite fields, we consider the setting of $n \times n$ matrix points with
finite field entries. We obtain exact formulas for matrix point counts on
elliptic curves and certain $K3$ surfaces for "supersingular" primes. These
exact formulas, which involve partitions of integers up to $n$, essentially
coincide with the expected value for the number of such points. Therefore, in
analogy with the Sato-Tate conjecture, it is natural to study the distribution
of the deviation from the expected values for all primes. We determine the
limiting distributions for elliptic curves and a family of $K3$ surfaces. For
non-CM elliptic curves with square-free conductor, our results are explicit.
