  We are concerned with the sign of the solutions of non-cooperative systems
when the parameter varies near a principal eigenvalue of the system. With this
aim we give precise estimates of the validity interval for the Antimaximum
Principle for an equation and an example. We apply these results to a
non-cooperative system. Finally a counterexample shows that our hypotheses are
necessary. The Maximum Principle remains true only for a restricted positive
cone.
