The main goal of this paper is to define and study of Rainville’s matrix polynomials. The integral
form, typical families of linear, bilinear and bilateral generating matrix functions are obtained
and gives expansions of these polynomials. The active differential operator on Rainville’s matrix
polynomials and their applications, some matrix recurrence relations are derived and the matrix
differential equation is obtained. Subsequently, Rodrigues formula are then developed for Rainville’s
matrix polynomials. Furthermore, we give the generating matrix functions for Rainville’s matrix
polynomials, involving the Horn’s matrix functions of two variables and hypergeometric matrix
functions of three variables.