This paper presents posterior Cram'er-Rao lower bounds (PCRLB) for extended target tracking (ETT) when the extent states of the targets are represented with random matrices. PCRLB recursions are derived for kinematic and extent states taking complicated expectations involving Wishart and inverse Wishart distributions. For some analytically intractable expectations, Monte Carlo integration is used. The bounds for the semi-major and minor axes of the extent ellipsoid are obtained as well as those for the extent matrix elements. The resulting bounds are compared on simulations with the performance of a state-of-the-art ETT algorithm employing random matrices for extent estimation.