The probability of rectangular unimodular matrices over $\F_q[x]$

Xiangqian Guo, Guangyu Yang

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…