  In this note, we compute the probability that a $k\times n$ matrix can be
extended to an $n\times n$ invertible matrix over $\F_q[x]$, which turns out to
be $(1-q^{k-n})(1-q^{k-1-n})...(1-q^{1-n})$. Connections with Dirichlet's
density theorem on the co-prime integers and its various generalizations are
also presented.
