### Explicit formulas and determinantal representations for $\eta$-skew-Hermitian solution to a system of quaternion matrix equations

#### Abstract

Some necessary and sufficient conditions for the existence of the $\eta$-skew-Hermitian solution quaternion matrix equations the system of matrix equations with $\eta$-skew-Hermicity,

\begin{align*}

&A_1X=C_1,~XB_1=C_2,\\

&A_2Y=C_3,~YB_2=C_4,\\

&X=-X^{\eta\ast},~Y=-Y^{\eta\ast},\\

&A_3Z=C_3,~ZB_3&=D_3,\\

&A_3XA_3^{\eta\ast}+B_3YB_3^{\eta\ast}=C_5,

\end{align*}

are established in this paper by using rank equalities of the coefficient matrices.

The general solutions to the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column determinants, we also give determinantal representation formulas of finding their exact solutions that are analogs of Cramer's rule.

A numerical example

is also given to demonstrate the main results.

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