  We define and study the $T\bar{T}$ deformation of a random matrix model,
showing a consistent definition requires the inclusion of both the perturbative
and non-perturbative solutions to the flow equation. The deformed model is well
defined for arbitrary values of the coupling, exhibiting a phase transition for
the critical value in which the spectrum complexifies. The transition is
between a single and a double-cut phase, typically third order and in the same
universality class as the Gross-Witten transition in lattice gauge theory. The
$T\bar{T}$ deformation of a double scaled model is more subtle and complicated,
and we are not able to give a compelling definition, although we discuss
obstacles and possible alternatives. Preliminary comparisons with finite
cut-off Jackiw-Teitelboim gravity are presented.
