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Explicit formulas and determinantal representation for η-skew-Hermitian solution to a system of quaternion matrix equations

Rehman Abdur (University of Engineering & Technology, Lahore, Pakistan), 1982pk@hotmail.com
Kyrchei Ivan (Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NASU, Lviv, Ukraine), kyrchei@online.ua
Ali Ilyas (University of Engineering & Technology, Lahore, Pakistan), ilyasali10@yahoo.com
Akram Muhammad (University of Engineering & Technology, Lahore, Pakistan), akramonline@hotmail.com
Shakoor Abdul (Khwaja Fareed University of Engineering & Information Technology, Rahim Yar Khan-Pakistan), ashakoor313@gmail.com

Some necessary and sufficient conditions for the existence of the η-skew-Hermitian solution quaternion matrix equations the system of matrix equations with η-skew-Hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X = -Xη*; Y=-Yη*, A3XAη*3 + B3YBη*3=C5, 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.

Keywords: Sylvester-type matrix equation, quaternion matrix, Moore-Penrose inverse, noncommutative determinant, Cramer’s Rule