  We study the smallest non-trivial matrix model that can be considered to be a
(toy) model of a black hole. The model consists of a pair of $2\times 2$
traceless hermitian matrices with a commutator squared potential and an $SU(2)$
gauge symmetry, plus an $SO(2)$ rotation symmetry. We show that using the
symmetries of the system, all but two of the variables can be separated. The
two variables that remain display chaos and a transition from chaos to
integrability when a parameter related to an $SO(2)$ angular momentum is tuned
to a critical value. We compute the Lyapunov exponents near this transition and
study the critical exponent of the Lyapunov exponents near the critical point.
We compare this transition to extremal rotating black holes.
