On matroids which have precisely one basis in common
A theorem concerning matroids is proved which - specialized to representable matroids - implies that given a non-singular quadratic \(n\times n\)-matrix \(A=(a_{ij})\) and some \(k\in \{1,...,n-1\}\) such that for any subset \(S\subseteq \{1,...,n\}\) of cardinality k different from \(\{\) 1,...,k\(\}\), the product \(\det ((a_{ij})_{i=1,...,k;j\in S})\cdot \det ((a_{ij})_{i=k+1,...,n;j\not\in S})\) in the Laplace expansion of A with respect to the first k and the last n-k rows vanishes, there exists some \(i\in \{k+1,...,n\}\) with \(a_{1i}=a_{2i}=...=a_{ki}=0\) or some \(i\in \{1,...,k\}\) with \(a_{k+1,i}=a_{k+2,i}=...=a_{ni}=0.\)
scientific article; zbMATH DE number 4083644