Final answer:
Clytemnestra uses sweet words and flattery to persuade Agamemnon to walk on the purple carpet, appealing to his vanity and status as a king.
Step-by-step explanation:
In the play Agamemnon by Aeschylus, Clytemnestra convinces Agamemnon to walk on the purple carpet by persuading him with sweet words. She appeals to his vanity and positions walking on the carpet as an honor fit for his royal status. Clytemnestra skillfully uses flattery and manipulates the situation to appear as though denying this grand gesture would be an act of false humility on Agamemnon's part.