We study low-rank matrix estimation for a generic inhomogeneous output channel through which the matrix is observed. This generalizes the commonly considered spiked matrix model with homogeneous noise to include for instance the dense degree-corrected stochastic block model. We adapt techniques used to study multi-species spin glasses to derive and rigorously prove an expression for the free energy of the problem in the large size limit, providing a framework to study the signal detection thresholds. We discuss an application of this framework to the degree corrected stochastic block models.