I present analytical results on the distribution of the largest eigenvalue for random matrices of the Cauchy type. Matrices belonging to this ensemble i) have a rotationally invariant weight, and ii) display a spectral density having support on the full real axis, which decays as a power law for x\to\pm\infty. The distribution exhibits a central regime that is governed by a scaling function (analogous to the Tracy-Widom distribution), flanked on both sides by large deviation tails. The corresponding rate functions are computed analytically using a mapping to a Coulomb gas system with constraints. The analytical results are corroborated by numerical simulations with excellent agreement.