Natural density of rectangular unimodular integer matrices

In this paper, we compute the natural density of the set of k×n integer matrices that can be extended to an invertible n×n matrix over the integers. As a corollary, we find the density of rectangular matrices with Hermite normal form Ok×(n-k)Ik. Connections with Cesàro's Theorem on the density of coprime integers and Quillen-Suslin's Theorem are also presented.