  We consider some combinatorial problems on matrix polynomials over finite
fields. Using results from control theory we give a proof of a result of
Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials
over a finite field. As an application of our results we give a new proof of a
theorem of Chen and Tseng which answers a question of Niederreiter on splitting
subspaces. We use our results to affirmatively resolve a conjecture on the
probability that a matrix polynomial is unimodular.
