  We equip a matrix algebra with a weighted infinitesimal unitary bialgebraic
structure, via a construction of a suitable coproduct. Furthermore, an
infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on
a matrix algebra. By exploring the relationship between weighted infinitesimal
bialgebras and pre-Lie algebras, we construct a pre-Lie algebraic structure and
then a new Lie algebraic structure on a matrix algebra. We also introduce the
weighted associative Yang-Baxter equations (AYBEs) and obtain the relationship
between solutions of weighted AYBEs and weighted infinitesimal unitary
bialgebras. We give a bijection between the solutions of the associative
Yang-Baxter equation of weight $\lambda$ and Rota-Baxter operators of weight
$-\lambda$ on matrix algebras. As a consequence, weighted quasitriangular
infinitesimal unitary bialgebras are constructed, which generalize the results
studied by Aguiar. Finally, We show that any weighted quasitriangular
infinitesimal unitary bialgebra can be made into a dendriform algebra.
