  In this paper, we establish interior $C^{1,\alpha}$ estimates for solutions
of the linearized Monge-Amp$\grave{e}$re equation
$$\mathcal{L}_{\phi}u:=\mathrm{tr}[\Phi D^2 u]=f,$$ where the density of the
Monge-Amp$\grave{e}$re measure $g:=\mathrm{det}D^2\phi$ satisfies a
$\mathrm{VMO}$-type condition and $\Phi:=(\mathrm{det}D^2\phi)(D^2\phi)^{-1}$
is the cofactor matrix of $D^2\phi$.
