Why does Don Pasquale agree to Ernesto and Norina’s marriage at the end of the opera?

Don Pasquale agrees to the marriage of Ernesto and Norina because he is desperate to rid himself of his supposed wife, "Sofronia," who has made his life miserable with her extravagant spending and defiant behavior.1

The situation is a ruse orchestrated by Dr. Malatesta. To convince the exhausted and humiliated Pasquale to consent to Ernesto’s marriage, Malatesta suggests that if Pasquale allows Ernesto to marry Norina, "Sofronia" will be so outraged by the prospect of sharing the house with her that she will agree to leave.1

Once Pasquale agrees to the union, Malatesta reveals that the entire marriage was a trick: "Sofronia" is actually Norina in disguise, and the initial marriage contract was invalid. Although Pasquale is initially stunned and calls them rascals, he is ultimately relieved to be free of his "wife" and happily grants his blessing to the young couple, acknowledging the folly of an old man attempting to marry.1