With the development of networking technology, the computing system has evolved towards the multi-tier paradigm gradually. However, challenges, such as multi-resource heterogeneity of devices, resource competition of services, and networked system dynamics, make it difficult to guarantee service-level agreement (SLA) for the applications. In this paper, we propose a multi-tier edge-cloud computing framework, EdgeMatrix, to maximize the throughput of the system while guaranteeing different SLA priorities. First, in order to reduce the impact of physical resource heterogeneity, EdgeMatrix introduces the Networked Multi-agent Actor-Critic (NMAC) algorithm to re-define physical resources with the same quality of service as logically isolated resource units and combinations, i.e., cells and channels. In addition, a multi-task mechanism is designed in EdgeMatrix to solve the problem of Joint Service Orchestration and Request Dispatch (JSORD) for matching the requests and services, which can significantly reduce the optimization runtime. For integrating above two algorithms, EdgeMatrix is designed with two time-scales, i.e., coordinating services and resources at the larger time-scale, and dispatching requests at the smaller time-scale. Realistic trace-based experiments proves that the overall throughput of EdgeMatrix is 36.7% better than that of the closest baseline, while the SLA priorities are guaranteed still.