On the Hesse-Muir formula for the determinant of the matrix \(A^{n-1}B^{2}\) (Q1957067)

Given a symmetric \(2\times 2\) matrix \(A\) and a \(2\times 2\) matrix \(B\) a formula for \(\det A\cdot(\det B)^2\) was presented by \textit{O.~Hesse} [J. Reine Angew. Math. 49, 243--264 (1855; ERAM 049.1314cj)]. The result is in terms of the determinant of a \(2\times 2\) matrix the \((i,j)\)-entry of which is itself a determinant, namely of a block matrix composed from \(A\), the transpose of the \(i\)-th row of \(B\), the \(j\)-th row of \(B\), and a zero entry. \textit{T.~Muir} [Sylv., Am. J. IV. 273--275 (1881; JFM 13.0124.01)] extended this to symmetric \(n\times n\) matrices \(A\) and \(n\times n\) matrices \(B\), and obtained a formula for \(\det(A^{n-1}B^2)\) in terms of the determinant of an \(n\times n\) matrix whose entries are themselves determinants arising in the way from above. The present short communication gives yet another generalisation by dropping the assumption that \(A\) should be symmetric. The proof is based on the idea to rewrite \(\det(A^{n-1}B^2)\) as \(\det (B(\text{adj}\,A)^t B^t)\) which allows to use a theorem from the work by \textit{V. V.~Prasolov} [Problems and theorems in linear algebra. Translations of Mathematical Monographs. 134. Providence, RI: AMS. (1994; Zbl 0803.15001)].