Summary: In this article, we study linear differential equations of higher-order whose coefficients are square matrices. The combinatorial method for computing the matrix powers and exponential is adopted. New formulas representing auxiliary results are obtained. This allows us to prove properties of a large class of linear matrix differential equations of higher-order, in particular results of T. M. Apostol [Am. Math. Mon. 82, 159–162 (1975; Zbl 0302.34001)] and I. I. Kolodner [J. Math. Anal. Appl. 52, 514–524 (1975; Zbl 0323.34009)] are recovered. Also illustrative examples and applications are presented.