  We explore the generalized $f(R,T)$ modified theory of gravity, where the
gravitational Lagrangian is a function of Ricci scalar $R$ and the trace of the
energy-momentum tensor $T$. We derive modified field equations to the linear
order of perturbations in the context of $f(R,T)$ model. We then investigate
the growth of perturbations in the context of $f(R,T)$ modified gravity.
Primary numerical investigations based on matter power spectra diagrams
indicates a structure growth suppression in $f(R,T)$ gravity, which exhibits
consistency with local measurements. Also, we notice that matter-geometry
interaction in $f(R,T)$ model would results in the specific feature named as
"matter acoustic oscillations" appeared in matter power spectra diagrams.
Moreover, we put constraints on the cosmological parameters of $f(R,T)$ model,
utilizing current observations, chiefly cosmic microwave background, weak
lensing, supernovae, baryon acoustic oscillations, and redshift-space
distortions data. Numerical results based on MCMC calculations imply that
$f(R,T)$ is a qualified theory of modified gravity in reconciling Planck CMB
data with local probes of large scale structures, by reporting lower values for
the structure growth parameter $\sigma_8$ compared to the standard model of
cosmology.
