This thesis studies a special multi-agent path finding problem known as the “Matrix Problem”. In the Matrix Problem, a set of agents, initially at rest in specific positions, must sequentially reach a common target without colliding. In this thesis, we model this as a set of synchronized collective motions which take place on the edges of a given graph, where the target is a particular vertex in such graph. Agents are of two types: the tasked ones that need their motion to pass through the target, and the untasked ones that can be moved to ease the motions of the tasked agents.