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.



Cayley-Hamilton revisited
Yes, this follows from known facts on matrix polynomials. There is a full characterization of spectral divisors of matrix polynomials in Gohberg, Lancaster, Rodman, *Matrix Polynomials*. They treat monic polynomials (i.e., $A\_k=1$), but this is not a restriction, since one can make a Möbius transform to enforce it unless $g(\lambda)\equiv 0$.


They introduce so-called *standard pairs* of a matrix polynomial $A(\lambda)$, which are in some sense a generalization of the companion matrix. Then they prove (Thm 3.12) that if a polynomial $Q(\lambda)$ is a right divisor of $P(\lambda)$ then the standard pair of $Q$ is a restriction of that of $P$; translating it to companion matrices, it would mean that the companion matrix of $Q$ can be obtained as a restriction of that of $P$ to an invariant subspace.


We apply that to $\lambda I - B$ (for which the "companion matrix" is $B$ itself) and $P(\lambda)=\sum\_{i=0}^k A\_i \lambda^i$. In particular, this means that the Jordan structure of $B$ is a substructure of that of the companion matrix of $P(\lambda)$, hence the algebraic multiplicities of the eigenvalues of $B$ are less or equal than those of $P(\lambda)$, that is, the characteristic polynomial of $B$ is a divisor of your $g(\lambda)$.


Not sure how much of this is understandable -- I must admit that book has not the reputation for being an easy-to-read one in the community. A more self-contained approach to these topics is in: I. Gohberg, M.A. Kaashoek and P. Lancaster,
*General theory of regular matrix polynomials and band Toeplitz operators*.


