  Motivated by the classical comatrix coalgebra, we introduce the concept of a
Newtonian comatrix coalgebra. We construct an infinitesimal unitary bialgebra
on a matrix algebra and a weighted infinitesimal unitary bialgebra on a
non-commutative polynomial algebra, via two constructions of suitable
coproducts. As a consequence, a Newtonian comatrix coalgebra is established.
Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar,
is constructed on a matrix algebra. By investigating the relationship between
weighted infinitesimal bialgebras and pre-Lie algebras, we erect respectively a
pre-Lie algebraic structure and further a new Lie algebraic structure on matrix
algebras. Finally, a pre-Lie algebraic structure and a Lie algebraic structure
on non-commutative polynomial algebras are also given.
