Local Multi-Trace Formulations (local MTF) are block-sparse boundary integral equations adapted to elliptic PDEs with piece-wise constant coefficients (typically multi-subdomain scattering problems) only recently introduced in [Hiptmair & Jerez-Hanckes, 2012]. In these formulations, transmission conditions are enforced by means of local operators, so that only adjacent subdomains communicate. We are interested here in the inverse of local multi-trace operators. We show that this inverse can be obtained in closed form for a model problem with three subdomains in the special case where the coefficients are homogeneous. We illustrate our findings with a numerical experiment that shows that discretizing the closed form inverse gives indeed and approximate inverse of the discretized local multi-trace operator.