In this paper, we derive Lyapunov type inequalities for a coupled system of Caputo fractional differential equations with boundary values. By constructing the associated Green's function and discussing its relevant properties, we provide two distinct proofs based on matrix spectral analysis and Perov's fixed point theorem, respectively. These analyses derive explicit necessary conditions on the coefficient functions for the existence of nontrivial solutions, extending the classical Lyapunov inequality theory to a coupled system framework involving differential and boundary coupling. The findings of this study are therefore valuable, offering new perspectives that enrich the literature.