We establish a connection between the Khatri-Rao and Tracy-Singh products introduced by Khatri and Rao (C.G. Khatri, C.R. Rao, Sankhya 30 (1968) 167-180) and Tracy and Singh (D.S. Tracy, R.P. Singh, Statistica Neerlandica 26 (1972) 143-157), respectively, and present further results including matrix equalities and inequalities involving the two products. Also, we give two statistical applications.