In this paper, we introduce and study a matrix analog of the five-parameter Mittag-Leffler function. We establish the absolute convergence of the series defining this function on the unit circle $|\varpi |=1$ under certain spectral conditions. Several fundamental properties are derived, including integral representations, derivative formulas, and differential recurrence relations. Furthermore, we obtain a variety of finite summation formulas for this matrix function and its related Fox-Wright matrix analog. Finally, we investigate the composition of the function with generalized fractional calculus operators introduced by Katugampola, deriving closed-form expressions for both fractional integrals and derivatives. The results presented here extend the theory of special matrix functions and contribute to the field of fractional calculus.