In the last two talks, we have studied a simplicial presentation of the Chow ring of a matroid given by Backman, Eur, and Simpson. This week, we will use this presentation to complete the proof that the Chow ring of a matroid is a Poincaré duality algebra. In particular, we will see that the Chow ring of a matroid is isomorphic to a quotient of the Chow ring of the braid fan by the annihilator of a homogeneous element. Since the Chow ring of the braid fan is a Poincaré duality algebra, it will follow that the Chow ring of a matroid is as well.