The question was sharp enough not to let Peter off, yet gentle enough not to let Peter go. Our Lord puts the same question to us in exactly the same spirit for exactly the same reason. Months earlier a woman whose tears bespoke as much as could ever be said anointed Jesus in public, witnessing to the watching world that she gloried in his priestly pardon and gladly submitted to his kingly claim. The purpose of the gospel is to quicken faith. The purpose of the church is to nurture faith.
Most people understand faith in terms of belief.
We should understand faith, rather, as the response we render our Lord when puts question after question to us. If we truly believe in him there's much that we shall do. It should have told the onlookers too. Our Lord was unfailingly concerned for the poor, as no doubt the Israelite woman herself was. Her deed couldn't tell onlookers how much she loved him, however, in that they lacked such love themselves, and lacking such love themselves were unable to recognize it in someone else.