This paper examines the cluster consensus problem of multi-agent systems on matrix-weighted switching networks. Necessary and/or sufficient conditions under which cluster consensus can be achieved are obtained and quantitative characterization of the steady-state of the cluster consensus are provided as well. Specifically, if the underlying network switches amongst finite number of networks, a necessary condition for cluster consensus of multi-agent system on switching matrix-weighted networks is firstly presented, it is shown that the steady-state of the system lies in the intersection of the null space of matrix-valued Laplacians corresponding to all switching networks. Second, if the underlying network switches amongst infinite number of networks, the matrix-weighted integral network is employed to provide sufficient conditions for cluster consensus and the quantitative characterization of the corresponding steady-state of the multi-agent system, using null space analysis of matrix-valued Laplacian related of integral network associated with the switching networks. In particular, conditions for the bipartite consensus under the matrix-weighted switching networks are examined. Simulation results are finally provided to demonstrate the theoretical analysis.