The probability of rectangular unimodular matrices over Fq[x]

In this note, we show that the probability that a uniformly random k × n matrix over Fq[x] can be extended to an n × n invertible matrix is (1-qk-n)(1-qk-1-n)⋯(1-q1-n). Connections with Dirichlet’s density theorem on co-prime integers and its various generalizations are also presented.