  In this paper, we consider a comprehensive investigation of the cosmological
model described by $f(R,T) = R + 2\lambda T$ (where $\lambda$ represents a free
parameter) in light of the most recent observational data. By constraining the
model using $Hubble$ and $Pantheon$ datasets, we determine its compatibility
with the observed behavior of the Universe. For this purpose, we adopt a
parametric form for the effective equation of state (EoS) parameter. This
parametric form allows us to describe the evolution of the EoS parameter with
respect to redshift and investigate its behavior during different cosmic
epochs. The analysis of the deceleration parameter reveals an accelerating
Universe with a present value of $q_0=-0.64^{+0.03}_{-0.03}$, indicating the
current phase of accelerated expansion. The transition redshift is found to be
$z_{tr}=0.53^{+0.04}_{-0.03}$, marking the epoch of transition from
deceleration to acceleration. We also analyze the evolution of important
cosmological parameters including density parameter, pressure, effective EoS,
and stability. These findings collectively demonstrate the viability of the
$f(R,T)$ cosmological model as a robust candidate capable of engendering the
requisite negative pressure, thereby efficiently propelling cosmic expansion.
Moreover, the undertaken stability analysis underscores the model's stability
within the broader cosmic landscape. By providing the best-fit values for the
coupling parameter $\lambda$, this approach motivates and encourages further
explorations into the extensive landscape of this model and its potential
applications across diverse realms of cosmology and astronomy.
