A double Sylvester determinant - Grinberg, Darij (Ars mathematica contemporanea, 2021)
Given two ?$\left( n+1\right) ×\left( n+1\right)$?-matrices ?$A$? and ?$B$? over a commutative ring, and some ?$k\in\left\{ 0,1,…,n\right\}$?, we consider the ?$\dbinom{n}{k}×\dbinom{n}{k}$?-matrix ?$W$? whose entries are ?$\left( k+1\right) ×\left( k+1\right)$?-minors of ?$A$? multiplied by corresponding ?$\left( k+1\right) ×\left( k+1\right)$?-minors of ?$B$?. Here we require the minors to use the last row and the last column (which is why we obtain an ?$\dbinom{n}{k}×\dbinom{n}{k}$?-matrix, not an ?$\dbinom{n+1}{k+1}×\dbinom{n+1}{k+1}$?-matrix). We prove that the determinant ?$\det W$? is a multiple of ?$\det A$? if the ?$\left( n+1,n+1\right)$?-th entry of ?$B$? is ?$0$?. Furthermore, if the ?$\left( n+1,n+1\right)$?-th entries of both ?$A$? and ?$B$? are ?$0$?, then ?$\det W$? is a multiple of ?$\left( \det A\right) \left( \det B\right)$?. This extends a previous result of Olver and the author.
Keywords: compound matrix; determinant; determinanta; polinomi; polynomials; sestavljena matrika; Sylvester's determinant; Sylvestrova determinanta