Through this article, we will discuss a new extension of the incomplete Wright hypergeometric matrix function by using the extended incomplete Pochhammer matrix symbol. First, we give a generalization of the extended incomplete Wright hypergeometric matrix function and state some integral equations and differential formulas about it. Next, we obtain some results about fractional calculus of these extended incomplete Wright hypergeometric matrix functions. Finally, we discuss an application of the extended incomplete Wright hypergeometric matrix function in the kinetic equations.