Am I allowed to empower Matriarch of War legend with "Kayle, Justified" second and third empower?

Yes — you can activate Kayle's Empower a second and third time, and Matriarch of War's trigger fires each time — but only the first one does anything to her.

Kayle's text "I can be Empowered up to three times" is a permission that overrides the normal "can't Empower something already Empowered" restriction (Rule 441.1.c.1), so the 2nd and 3rd Empower activations are legal.

Each activation is an "empower something else" event, so Matriarch of War's trigger goes on the Chain every time. But since Matriarch is already Empowered after the first one, the 2nd and 3rd triggers do nothing when they resolve — her own text says "I become Empowered if I'm not already," and Empowered is a binary state (Rule 441.1.a–c). She doesn't gain extra Empower stacks; only Kayle tracks her count.