The main aim of this paper is to define a new family of special matrix functions, say, -Hypergeometric, -Bessel, -modified Bessel and -Tricomi matrix functions, where is a positive integer. The convergence properties of the -Hypergeometric matrix function are established and the matrix differential equation is constructed. The differential properties, integral representations, integrals and various particular cases on the $p$-Bessel matrix function have been obtained. The -modified Bessel matrix functions, and some properties of this function are investigated. Finally, the -Tricomi matrix function and several results of the function are defined and studied.