Some novel results on a classical system of matrix equations over the dual quaternion algebra

We employ the M-P inverses and ranks of quaternion matrices to establish the necessary and sufficient conditions for the solvability of a system of the dual quaternion matrix equations (AX, XC) = (B, D), along with providing an expression for its general solution. In addition, we investigate the solutions to the dual quaternion matrix equations AX = B and XC = D, including eta-Hermitian solutions. Serving as applications, we design a scheme for encrypting and decrypting color images based on this system of dual quaternion matrix equation, and experimental results show that the scheme is highly feasible.