  Classical hypergeometric functions are well-known to play an important role
in arithmetic algebraic geometry. These functions offer solutions to ordinary
differential equations, and special cases of such solutions are periods of
Picard-Fuchs varieties of Calabi-Yau type. Gauss' $_2F_1$ includes the
celebrated case of elliptic curves through the theory of elliptic functions. In
the 80s, Greene defined finite field hypergeometric functions that can be used
to enumerate the number of finite field points on such varieties. We extend
some of these results to count finite field ``matrix points." For example, for
every $n\geq 1,$ we consider the matrix elliptic curves $$ B^2 = A(A-I_n)(A-a
I_n), $$ where $(A,B)$ are commuting $n\times n$ matrices over a finite field
$\mathbb{F}_q$ and $a\neq 0,1$ is fixed. Our formulas are assembled from
Greene's hypergeometric functions and $q$-multinomial coefficients. We use
these formulas to prove Sato-Tate distributions for the error terms for matrix
point counts for these curves and some families of $K3$ surfaces.
