  We give combinatorial proofs of two multivariate Cayley--Hamilton type
theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving
$2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed
discriminant, a matrix function which has generated a lot of interest in recent
times. Recently, the Cayley--Hamilton theorem for mixed discriminants was
proved by Bapat and Roy (Comb. Math. and Comb. Comp., 2017). We prove a
Phillips-type generalization of the Bapat--Roy theorem involving $2nk$
matrices, where $n$ is the size of the matrices, among which $nk$ commute
pairwise. Our proofs generalize the univariate proof of Straubing (Disc. Math.,
1983) for the original Cayley--Hamilton theorem in a nontrivial way, and
involve decorated permutations and decorated paths.
