  Characterizing the entanglement of matrix degrees of freedom is essential for
understanding the holographic emergence of spacetime. The Quantum Hall Matrix
Model is a gauged $U(N)$ matrix quantum mechanics with two matrices whose
ground state is known exactly and describes an emergent spatial disk with
incompressible bulk dynamics. We define and compute an entanglement entropy in
the ground state associated to a cut through the disk. There are two
contributions. A collective field describing the eigenvalues of one of the
matrices gives a gauge-invariant chiral boundary mode leading to an expected
logarithmic entanglement entropy. Further, the cut through the bulk splits
certain 'off-diagonal' matrix elements that must be duplicated and associated
to both sides of the cut. Sewing these duplicated modes together in a
gauge-invariant way leads to a bulk 'area law' contribution to the entanglement
entropy. All of these entropies are regularized by finite $N$.
