  We study an elliptic system coupled by Monge-Amp\`{e}re equations:
  \begin{center}
  $\left\{
  \begin{array}{ll}
  det~D^{2}u_{1}={(-u_{2})}^\alpha, & \hbox{in $\Omega,$}
  det~D^{2}u_{2}={(-u_{1})}^\beta, & \hbox{in $\Omega,$}
  u_{1}<0, u_{2}<0,& \hbox{in $\Omega,$}
  u_{1}=u_{2}=0, & \hbox{on $ \partial \Omega,$}
  \end{array}
  \right.$
  \end{center} here $\Omega$~is a smooth, bounded and strictly convex domain
in~$\mathbb{R}^{N}$,~$N\geq2,~\alpha >0,~\beta >0$. When $\Omega$ is the unit
ball in $\mathbb{R}^{N}$, we use index theory of fixed points for completely
continuous operators to get existence,
  uniqueness results and nonexistence of radial convex solutions under some
corresponding assumptions on $\alpha,\beta$. When $\alpha>0$, $\beta>0$ and
$\alpha\beta=N^2$
  we also study a corresponding eigenvalue problem in more general domains.
