Abstract
In this paper, some important objectives have been achieved, which are as follows: First, we present a method of the inverse for a class of non-singular block matrices and some associated properties. Also, the accuracy of a new method is verified with some illustrated examples by applying the MATLAB lines. Second, applying a class of block matrices, we give the exact solution for fractional matrix differential equation systems using the Laplace fractional transformation method. Finally, illustrative examples and individual cases are also presented and discussed to demonstrate our new approach.