the (r,s)-symmetric and (r,s)-skew symmetric solutions of the pair of matrix equations a1xb1 = c1 and a2xb2 = c2

let $rin textbf{c}^{mtimes m}$ and $sin textbf{c}^{ntimes n}$ be nontrivial involution matrices; i.e., $r=r^{-1}neq pm~i$ and $s=s^{-1}neq pm~i$. an $mtimes n$ complex matrix $a$ is said to be an $(r, s)$-symmetric ($(r, s)$-skew symmetric) matrix if $ras =a$ ($ ras =-a$). the $(r, s)$-symmetric and $(r, s)$-skew symmetric matrices have a number of special properties and widely used in engineering and scientific computating. here, we introduce the necessary and sufficient conditions for the solvability of the pair of matrix equations $a_{1}xb_{1}=c_{1}$ and $a_{2}xb_{2}=c_{2}$, over $(r, s)$-symmetric and $(r, s)$-skew symmetric matrices, and give the general expressions of the solutions for the solvable cases. finally, we give necessary and sufficient conditions for the existence of $(r, s)$-symmetric and $(r, s)$-skew symmetric solutions and representations of these solutions to the pair of matrix equations in some special cases.