In surveillance applications, the extent states and measurements of extended targets received by sensors are time-varying. In this paper, we propose a joint tracking and classification (JTC) method for single extended target under the presence of clutter and detection uncertainty. The extent state is modeled as elliptic shape via random matrix model (RMM), and is used as the feature for target classification. To adapt to the time-varying conditions of an extended target, the RMM proposed by Lan et al. is used. Besides, the RMM is integrated into Bernoulli filter to detect an extended target with clutter and detection uncertainty. The resulting method is called joint tracking and classification Gaussian inverse Wishart Bernoulli (JTC-GIW-Ber) filter, and the closed expressions for JTC-GIW-Ber filter recursions are derived under the necessary assumptions and approximations. Comprehensive simulations are carried out to test the performance, and the results demonstrate that the proposed JTC-GIW-Ber filter not only outperforms the JTC-GIW probability hypothesis density (JTC-GIW-PHD) filter and the GIW-Ber filter in extent state estimation, but also outperforms the JTC-GIW-PHD filter in target classification. INDEX TERMS Bernoulli filter, joint tracking and classification, random matrix model, single extended target.