A formula for the determinant of a sum of matrices
Given a square matrix \(x\) over a commutative ring, define functions \(\Lambda_ i(x)\) by the equality \[ \det (1-tx)=1-t\Lambda_ 1(x)+t^ 2\Lambda^ 2_ 2(x)+\ldots+(-1)^ nt^ n\Lambda^ n_ n(x)+\ldots . \] The following is the main theorem: Let \(X_ 1,...,X_ k\) be square matrices of the same order and \(n\geq 1\). Then \(\Lambda_ n(X_ 1+\ldots+X_ k)=\sum_{m}sgn(m)\Lambda (m)\) where the sum is extended to the \(k^ n\) monomials of length n of primitive circular words on \(X_ 1,\ldots,X_ k\). The Hamilton-Cayley theorem is deduced from this formula.
scientific article; zbMATH DE number 4021349