This paper addresses a distributed optimization problem for first-order hybrid multi-agent systems on matrix-weighted networks, and proposes a novel distributed optimization control algorithm, in which a sampled control method is utilized in the continuous-time subsystem. Under the proposed optimal control protocol, an algebraic condition for achieving optimization consensus in the hybrid systems is established based on matrix theory, the Lyapunov stability theory, and inequality techniques, and further an algebraic graph condition is derived. Specifically, if the sampling period for the hybrid system satisfies certain condition and the local cost function of each agent is strongly convex, the system can achieve consensus at the global optimal solution when the null space of the Laplacian matrix spans the consensus subspace or the matrix-weighted graph contains a positive spanning tree. Finally, the effectiveness of the proposed algorithm is verified by numerical simulation.