Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application

In this paper, we establish the maximal and minimal ranks of the solution to the consistent system of quaternion matrix equations A1X=C1,A2X=C2,A3XB3=C3 and A4XB4=C4, which was investigated recently by Wang [Q.W. Wang, The general solution to a system of real quaternion matrix equations, Comput. Math. Appl. 49 (2005) 665-675]. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition for the invariance of the rank of the general solution to the system mentioned above is presented. Some previous known results can be regarded as the special cases of this paper.