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In this paper, we first establish a stability result for a class of linear switching systems involving Kronecker product. The problem is intriguing in that the system matrix does not have to be Hurwitz in any time instant. We have established the main result by a combination of the Lyapunov stability analysis and a generalized Barbalat's Lemma applicable to piecewise continuous linear systems. As applications of this stability result, we study both the leaderless consensus problem and the leader-following consensus problem for general marginally stable linear multi-agent systems under switching network topology. In contrast with many existing results, our result only assume that the dynamic graph is uniformly connected.