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In this paper, we study a simultaneous stabilization and synchronization (SSS) problem for one class of linear multi-agent systems with non-identical agent dynamics and switching information-exchange topology. We show that a single control loop can be designed for each agent to enable some internal agent states to be stabilized while some other external states to be synchronized. We design a distributed control law based on local measurements and information exchanged from neighboring agents to enable SSS. The sufficient conditions to achieve SSS are obtained by properly designing a SSS reference system, followed by specific approaches of designing the control gain matrices. An example and simulation results are provided to demonstrate the effectiveness of the proposed method.