  The $SU(N)$ Yang-Mills matrix model admits self-dual and anti-self-dual
instantons. When coupled to $N_f$ flavors of massless quarks, the Euclidean
Dirac equation in an instanton background has $n_+$ positive and $n_-$ negative
chirality zero modes. We show that the index $(n_+ - n_-)$ is equal to a
suitably defined instanton charge. Further, we show that the path integral
measure is not invariant under a chiral rotation, and relate the non-invariance
of the measure to the index of the Dirac operator. Axial symmetry is broken
anomalously, with the residual symmetry being a finite group. For $N_f$
fundamental fermions, this residual symmetry is $\mathbb{Z}_{2N_f}$, whereas
for adjoint quarks it is $\mathbb{Z}_{4N_f}$.
