  We introduce a matrix model for noncommutative gravity, based on the gauge
group $U(2) \otimes U(2)$. The vierbein is encoded in a matrix $Y_{\mu}$,
having values in the coset space $U(4)/ (U(2) \otimes U(2))$, while the spin
connection is encoded in a matrix $X_\mu$, having values in $U(2) \otimes
U(2)$. We show how to recover the Einstein equations from the $\theta \to 0$
limit of the matrix model equations of motion. We stress the necessity of a
metric tensor, which is a covariant representation of the gauge group in order
to set up a consistent second order formalism. We finally define noncommutative
gravitational instantons as generated by $U(2) \otimes U(2)$ valued
quasi-unitary operators acting on the background of the Matrix model. Some of
these solutions have naturally self-dual or anti-self-dual spin connections.
