  Let K be a field and let M_n(K) denote the space of n x n matrices with
entries in K. Let M be a subspace of M_n(K) of dimension d with the property
that there are elements in M with non-zero determinant. Given a basis of M, we
define the determinantal polynomial P_{M} of M with respect to the basis. It is
a homogeneous polynomial of degree n in d indeterminates that gives the
determinant of any element of M by evaluation in K^d.
  This paper investigates the interrelationship of M and P_{M}. We confine
ourselves to finite fields K, where we can obtain useful information by
applying the Lang-Weil theorem on the number of zeros of absolutely irreducible
polynomials. A combination of Chevalley's theorem on the zeros of polynomials
in several variables and the Lang-Weil theorem leads to theorems about the
characteristic polynomials of elements of M when n is a prime. We also draw
attention to cases when the elements of M with non-zero determinant are a
proper subspace, and provide non-trivial examples of this phenomenon.
