Matrix Differential Equation (MDE) is a method to represent linear differential equations in terms of vectors. As you formulate MDE it is easy to solve it algebraically or numerically.
It is easy to get solution for algebraic equations, but in order to include all system dynamics you need to solve N-Linear differential equations. In order to get clear picture of the system analysis, you separate algebraic part and find its separate solution. In matrix differential equations you bifurcate the state vector into different components and then get individual solutions through ADE( Algebraic differential equation).