  We extend the idea of unimodular gravity to the modified $f(R,T)$ theories. A
new class of cosmological solutions, that the unimodular constraint on the
metric imposes on the $f(R,T)$ theories, are studied. This extension is done in
both Jordan and Einstein frames. We show that while the Lagrange multiplier
(that imposes the unimodular constraint on the action) depends on the cosmic
time in Jordan frame and therefore, can act as an evolving scalar field in the
universe history, in the Einstein frame it acts as a cosmological constant.
Then a general reconstruction method is used to realize an explicit form of the
unimodular $f(R,T)$ corresponding to a given cosmological solution. By adopting
a specific form of $f(R,T)$, the issue of cosmological inflation is studied in
this setup. To see the observational viability of this model, a numerical
analysis on the model parameter space is done in the background of Planck2015
observational data.
