In this talk we discuss 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. This talk is based on a joint work with Samrith Ram.
