  The paper extends basic Einstein--Hilbert action by adding a newly proposed
invariant constructed from a specific contraction between the Einstein tensor
and the energy momentum tensor, encoding a non--minimal coupling between the
space--time geometry and the matter fields. The fundamental Einstein--Hilbert
action is extended by considering a generic function ${f}(R,G_{\mu \nu}T^{\mu
\nu})$ which is further decomposed into its main constituents, a geometric
component which depends on the scalar curvature, and a second element embedding
the interplay between geometry and matter fields. Specific cosmological models
are established at the level of background dynamics, based on particular
couplings between the matter energy--momentum tensor and the Einstein tensor.
After deducing the resulting field equations, the physical aspects for the
cosmological model are investigated by employing a dynamical system analysis
for various coupling functions. The investigation showed that the present model
is compatible with different epochs in the evolution of our Universe, possible
explaining various late time historical stages.
