Natural density of rectangular unimodular integer matrices

Abstract

An integer matrix of size k×n, k≤n, is called unimodular if it can be extended to an n×n invertible matrix. The natural density of unimodular k×n matrices, which may be explained as the “probability" of a random k×n integer matrix to be unimodular, is determined in this paper using the Riemann's zeta function. The present result is a generalization of a classical result due to Cesáro and is also related to Quillen-Suslin's Theorem .