Abstract: In this paper we study a matrix equation AX+BX=C(I)over an arbitrary skewfield,and give a consistency criterion of(I)and an explicit expression of general solutions of(I).Aconvenient,simple and practical method of solving(I)is also given.As a particular case,we alsogive a simple method of finding a system of fundamental solutions of a homogeneous system of rightlinear equations over a skew field.

Key words: matrix cquation over a skew ficld, fundamental system of solutions, basic solution matrix, subdirect product, homogencous system of right linear ejations