  Determinants of the second-rank tensors stand useful in forming generally
invariant terms as in the case of the volume element of the gravitational
actions. Here, we extend the action of the matter fields by an arbitrary
function $f(D)$ of the determinants of their energy-momentum, and the metric,
$D=|\textbf{det}.T|/|\textbf{det}.g|$. We derive the gravitational field
equations and examine the nonlinear terms induced by the determinant,
specifically, the inverse of the energy-momentum tensor. We also show that
these extensions require a nonzero stress-energy tensor for the vacuum. We
propose a scale-free model, $f(D)=\lambda D^{1/4}$, and show how it induces the
familiar invariant terms formed by the trace of the energy-momentum tensor by
expanding the action around the stress-energy of the vacuum. We study the
hydrostatic equilibrium equations for a neutron star by providing relevant
values of the dimensionless constant $\lambda$. We show that the differences
from the predictions of general relativity, in the mass-radius relations, which
are sensitive to the equations of state, are conspicuous for $\lambda \sim
-10^{-2}$. We also show that the model does not affect the predictions on the
primordial nucleosynthesis when it is applied to the early radiation era. This
novel and unfamiliar type of gravity-matter coupling can lead to a rich
phenomenology in gravitational physics.
