Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let $g = \det(\lambda^kA_k+\cdots+\lambda A_1+A_0)\in K[\lambda]$.

Question: is it true that, if $B$ is a solution of (1) ($f(B)=0_n$), then $g(B)=0$ ?

i) In 1884, Sylvester seemed to think that it is true, but, without proof (as often with him).

ii) This result is "proved" in: K. Kanwar. A generalization of the Cayley-Hamilton Theorem. Advances in PURE mathematics.2013. Yet, the proof is obviously false !! Recall that this journal has already been the subject of a scandal: http://boingboing.net/2012/10/19/math-journal-accepts-computer.html So the problem seems not settled.

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