Matvei Ilyich Kolyazin is a successful relative of Arkady's, descended from his grandmother's side, and for a time he took care of Nikolai and Pavel Petrovich. He is a progressive statesman who has a high opinion of himself, but knows how to hide it.
The narrator tells us, "He was a skilful courtier, extremely sly, and that was all; he had no aptitude for handling public affairs, and no intelligence – though he knew how to manage his own business successfully: no one could get the better of him in this domain, and of course that is what really matters" (12.1).
When Bazarov and Arkady go to visit him in town, he takes Arkady under his wing and invites him to a ball being held in his honor. He is polite to Bazarov, but clearly is not impressed by the poorly dressed young man. It is at Matvei Ilyich's ball that Arkady first meets Anna Sergeyevna Odintsov.
At the end of the novel, we learn that Pavel Petrovich has become a well-respected figure abroad. One of the measures of his success is that Matvei Ilyich makes a point of visiting him when he travels.