Building matrices with prescribed size and number of invertible submatrices

Edward S. T. Fan, Tony W. H. Wong

Given an ordered triple of positive integers $(n,r,b)$, where $1\le b\le \left(\genfrac{}{}{0ex}{}{n}{r}\right)$, does there exist a matrix of size $r\times n$ with exactly $b$ invertible submatrices of si…