Large deviations of the maximum eigenvalue to the left of the expected value are investigated for the Gaussian and Wishart random matrices. Universal rate functions can be computed analytically with a Coulomb gas approach and numerical simulations are in good agreement with the theoretical predictions. In contrast with the case of independent random variables, the exponential decay of the probability of extreme events follows a power N² and not N due to the peculiar level repulsion.

PACS numbers: 02.50.--r, 02.10.Yn, 05.90.+m