  In this paper we consider the Matsumoto metric
$F=\frac{\alpha^2}{\alpha-\beta}$, on the three dimensional real vector space
and obtain the partial differential equations that characterize the minimal
surfaces which are graphs of smooth functions and then we prove that plane is
the only such surface. We also obtain the partial differential equation that
characterizes the minimal translation surfaces and show that again plane is the
only such surface.
