  We introduce a dually-weighted multi-matrix model that for a suitable choice
of weights reproduce two-dimensional Causal Dynamical Triangulations (CDT)
coupled to the Ising model. When Ising degrees of freedom are removed, this
model corresponds to the CDT-matrix model introduced by Benedetti and Henson
[Phys. Lett. B 678, 222 (2009)]. We present exact as well as approximate
results for the Gaussian averages of characters of a Hermitian matrix $A$ and
$A^2$ for a given representation and establish the present limitations that
prevent us to solve the model analytically. This sets the stage for the
formulation of more sophisticated matter models coupled to two-dimensional CDT
as dually weighted multi-matrix models providing a complementary view to the
standard simplicial formulation of CDT-matter models.
