| Summary: | The system of two-sided quaternion matrix equations with η-Hermicity,
A1XA1η* = C1
A
1
X
A
1
η
*
=
C
1
$ {\mathbf{A}}_1\mathbf{X}{\mathbf{A}}_1^{\eta \mathrm{*}}={\mathbf{C}}_1$
,
A2XA2η* = C2
A
2
X
A
2
η
*
=
C
2
$ {\mathbf{A}}_2\mathbf{X}{\mathbf{A}}_2^{\eta \mathrm{*}}={\mathbf{C}}_2$
is considered in the paper. Using noncommutative row-column determinants previously introduced by the author, determinantal representations (analogs of Cramer’s rule) of a general solution to the system are obtained. As special cases, Cramer’s rules for an η-Hermitian solution when
C1 = Cη*1
C
1
=
C
1
η
*
$ {\mathbf{C}}_1={\mathbf{C}}_1^{\eta \mathrm{*}}$
and
C2 = Cη*2
C
2
=
C
2
η
*
$ {\mathbf{C}}_2={\mathbf{C}}_2^{\eta \mathrm{*}}$
and for an η-skew-Hermitian solution when
C1 = −Cη*1
C
1
=
-
C
1
η
*
$ {\mathbf{C}}_1=-{\mathbf{C}}_1^{\eta \mathrm{*}}$
and
C2 = −Cη*2
C
2
=
-
C
2
η
*
$ {\mathbf{C}}_2=-{\mathbf{C}}_2^{\eta \mathrm{*}}$
are also explored.
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