The probability of rectangular unimodular matrices over ${\text{F}}_q[x]$

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