This　paper　investigates　the　asymptotic　stability　of　a　class　of　switched　linear　time-invariant　discrete-time　systems.　Without　the　restriction　of　switching　signals,　a　novel　result　is　first　proposed　to　guarantee　uniform　global　asymptotic　stability.　Then,　it　is　applied　to　studying　some　consensus　problems　of　linear　multi-agent　systems　under　switching　network　topology.　The　proposed　result　modifies　some　insufficient　condition　assumed　in　a　related　paper　and　establishes　the　exponentially　fast　convergence　rate　of　the　multi-agent　system.　An　interesting　example　illustrates　the　effectiveness　of　the　derived　results.