Ranks of the common solution to six quaternion matrix equations

A new expression is established for the common solution to six classical linear quaternion matrix equations A ₁ X = C ₁, XB ₁ = C ₃, A ₂ X = C ₂, XB ₂ = C ₄, A ₃ XB ₃ = C ₅, A ₄ XB ₄ = C ₆ which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721??32 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.