  In this note, we prove a uniqueness result, up to a positive multiplicative
constant, for nontrivial convex solutions to a system of Monge-Amp\`ere
equations \begin{equation*} \left\{ \begin{alignedat}{2} \det D^2 u~& = \gamma
|v|^p~&&\text{in} ~ \Omega, \\\ \det D^2 v~& = \mu |u|^{n^2/p}~&&\text{in} ~
\Omega, \\\ u=v &= 0~&&\text{on}~ \partial\Omega \end{alignedat} \right.
\end{equation*} on bounded, smooth and uniformly convex domains $\Omega\subset
R^n$ provided that $p$ is close to $n\geq 2$. When $p=n$, we show that the
uniqueness holds for general bounded convex domains $\Omega\subset R^n$.
