The probability of rectangular unimodular matrices over Fq[x]

In this note, we show that the probability that a uniformly random k x n matrix over F-q[x] can be extended to an n x n invertible matrix is (1 - q(k-n)) (1 - q(k-1-n)) ... (1-q(1-n)). Connections with Dirichlet's density theorem on co-prime integers and its various generalizations are also presented. (C) 2012 Elsevier Inc. All rights reserved.