The solvability and general solutions to two dual quaternion matrix equations with applications

In this paper, we consider the dual quaternion matrix equations AX+BYC=D and XA+BYC=D. We derive the necessary and sufficient conditions for their solvability, which involve both matrix equalities and rank equalities, as well as the general solution expressions when they are consistent. Additionally, we present several corollaries for special cases of the first equation when its coefficient matrices take particular forms. The paper concludes with a numerical example that verifies the main theoretical results. Moreover, two distinct encryption and decryption frameworks for colour images were introduced, and their feasibility and effectiveness were demonstrated through a concrete example using the first framework with two selected colour images.