In this paper, we propose a necessary and sufficient condition for the solvability to a system of matrix equations over the commutative quaternion ring, and establish an expression of its general solution when it is solvable. We also present an algorithm for finding an approximate solution to the system when it is inconsistent. Finally, we give an example to illustrate the main results of this paper.

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 ?-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.

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics and others. Due to its application in control theory, the matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C over the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of the dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for the dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.