The dual fuzzy matrix equations: Extended solution, algebraic solution and solution

In this paper, we propose a direct method to solve the dual fuzzy matrix equation of the form $\mathbf{A}\tilde{\mathbf{X}}+\tilde{\mathbf{B}}=\mathbf{C}\tilde{\mathbf{X}}+\tilde{\mathbf{D}}$ with $\mathbf{A}$, $\mathbf{C}$ matrices of crisp coefficients and $\tilde{\mathbf{B}}$, $\tilde{\mathbf{D}}$ fuzzy number matrices. Extended solution and algebraic solution of the dual fuzzy matrix equations are defined and the relationship between them is investigated. This article focuses on the algebraic solution and a necessary and sufficient condition for the unique algebraic solution existence is given. By algebraic methods we not need to transform a dual fuzzy matrix equation into two crisp matrix equations to solve. In addition, the general dual fuzzy matrix equations and dual fuzzy linear systems are investigated based on the generalized inverses of the matrices. Especially, the solution formula and calculation method of the dual fuzzy matrix equation with triangular fuzzy number matrices are given and discussed. The effectiveness of the proposed method is illustrated with examples.