Bulletin of the
Korean Mathematical Society BKMS

Assume that $X$, partitioned into $2\times2$ block form, is a solution of the system of quaternion matrix equations $A_{1} XB_{1}=C_{1},A_{2}XB_{2}=C_{2}.$ We in this paper give the maximal and minimal ranks of the submatrices in $X,$ and establish necessary and sufficient conditions for the submatrices to be zero, unique as well as independent. As applications, we consider the common inner inverse $G,$ partitioned into $2\times2$ block form, of two quaternion matrices $M$ and $N.$ We present the formulas of the maximal and minimal ranks of the submatrices of $G,$ and describe the properties of the submatrices of $G$ as well. The findings of this paper generalize some known results in the literature.

Keywords: matrix equation, minimal rank, maximal rank, generalized inverse, quaternion matrix, partitioned matrix

MSC numbers: 15A03, 15A09, 15A24, 11R52