  A detailed derivation of $3+1$ dimensional induced or emergent gravity in the
IKKT matrix model at one loop is given, as announced in [1]. The mechanism
requires a brane configuration with structure ${\cal M}^{3,1}\times {\cal K}
\subset {\mathbb R}^{9,1}$, where ${\cal M}^{3,1}$ is the noncommutative
space-time brane, and ${\cal K}$ are compact fuzzy extra dimensions embedded in
target space. The 3+1-dimensional Einstein-Hilbert action then arises in the
one loop effective action of the maximally supersymmetric IIB or IKKT matrix
model, with effective Newton constant determined by the Kaluza-Klein scale of
${\cal K}$. At weak coupling, all physical modes are confined to the brane,
leading to $3+1$-dimensional low-energy physics. The Einstein-Hilbert action
can be interpreted as interaction of ${\cal K}$ with the space-time brane via
IIB supergravity. The vacuum energy does not act as cosmological constant, but
stabilizes the brane structure at one loop.
