The decoupling of linear, time-invariant, multivariable systems into single-input, multiple-output subsystems is considered and a theorem by Falb and Wolovich completely solving a special case of this problem is extended to provide a strong necessary condition for decoupling. Unlike other results establishing necessary and sufficient conditions for this problem, the test developed herein is easily applied and does not require extensive constructions. Finally it is shown that this result extends easily to the more general case of decoupling a linear system into multi-variable subsystems.