  We establish a new class of examples of the multivariate Bateman-Horn
conjecture by using tools from dynamics. These cases include the determinant
polynomial on the space of $n\times n$ matrices, the Pfaffian on the space of
skew-symmetric $2n\times 2n$ matrices, and the determinant polynomial on the
space of symmetric $n\times n$ matrices. In particular, let $(V,F)$ be any pair
among the following: $(\textrm{Mat}_n, \det)$,
$(\textrm{Skew}_{2n},\textrm{Pff})$, and $(\textrm{Sym}_n, \det).$ We then
obtain an asymptotic for $$\pi_{V,F}(T)= \#\{v\in V: \max(|v_i|)\leq T, F(v)
\text{ is prime}\},$$ that matches the Bateman-Horn prediction. The key
ingredients of our proof are an asymptotic count for integral points on the
level sets of $F$ given by Linnik equidistribution, a geometric approximation
of the box by cones, and an upper bound sieve to bound the number of prime
values missed by the approximation. In the case of the determinant polynomial
on symmetric matrices, we must also use the Siegel mass formula to compute the
product of local densities for the main term.
