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This paper proposes a novel free-weighting matrix (FWM) approach for the event-triggered (ET) consensus and formation control of multi-agent systems over a directed graph. A new FWM equality for the consensus error is proposed. This FWM is applied to attain coherent behavior of systems via a fully distributed ET approach. The proposed approach is also extended for ET formation control to attain a specific pattern of agents. Furthermore, an FWM-based consensus controller design with practical ET cooperative control is considered to address consensus and formation control problems by using the in-degree information of a node rather than the algebraic connectivity of a graph or adaptation of parameters. Additionally, the controller gain is formulated using a more general transformation compared to that used in classical approaches. Moreover, the proposed ET scheme for the designed controller is applied on the transmission side to preserve computational resources by eliminating the Zeno behavior. In contrast to the existing methods, the proposed approach (i) provides a fully distributed protocol without using central information of network or adaptive gain; (ii) provides a practical ET mechanism to assure effective bandwidth and fewer energy consumption; and (iii) can be applied to the ET formation problem for mobile agents. Simulations involving six aerial vehicles are performed to demonstrate the efficacy of the proposed cooperative control strategy. [ABSTRACT FROM AUTHOR]