  In this paper we suggest generalizations of elliptic integrable tops to
matrix-valued variables. Our consideration is based on $R$-matrix description
which provides Lax pairs in terms of quantum and classical $R$-matrices. First,
we prove that for relativistic (and non-relativistic) tops such Lax pairs with
spectral parameter follow from the associative Yang-Baxter equation and its
degenerations. Then we proceed to matrix extensions of the models and find out
that some additional constraints are required for their construction. We
describe a matrix version of ${\mathbb Z}_2$ reduced elliptic top and verify
that the latter constraints are fulfilled in this case. The construction of
matrix extensions is naturally generalized to the monodromy preserving
equation. In this way we get matrix extensions of the Painlev\'e VI equation
and its multidimensional analogues written in the form of non-autonomous
elliptic tops. Finally, it is mentioned that the matrix valued variables can be
replaced by elements of noncommutative associative algebra. In the end of the
paper we also describe special elliptic Gaudin models which can be considered
as matrix extensions of the (${\mathbb Z}_2$ reduced) elliptic top.
