  The Cayley-Hamilton-Newton theorem - which underlies the Newton identities
and the Cayley-Hamilton identity - is reviewed, first, for the classical
matrices with commuting entries, second, for two q-matrix algebras, the
RTT-algebra and the RLRL-algebra. The Cayley-Hamilton-Newton identities for
these q-algebras are related by the factorization map. A class of algebras
M(R,F) is presented. The algebras M(R,F) include the RTT-algebra and the
RLRL-algebra as particular cases. The algebra M(R,F) is defined by a pair of
compatible matrices R and F. The Cayley-Hamilton-Newton theorem for the
algebras M(R,F) is stated. A nontrivial example of a compatible pair is given.
