Dual orthogonal tripotent matrices

In this paper, we study dual orthogonal tripotent matrices, defined by ${\widehat{A}}^3=\widehat{A}={\widehat{A}}^{\ast }$, and examine their fundamental algebraic properties. Additionally, we establish several characterizations of this class of matrices using matrix averages involving $\widehat{A}$, ${\widehat{A}}_e$, ${\widehat{A}}^{\ast }$, $\widehat{A}\text{sp}N$, as well as integer powers of products such as $\widehat{A}{\widehat{A}}^{\ast }$ and ${\widehat{A}}^{\ast }\widehat{A}$. These results enrich the theory of dual generalized matrix classes and reveal new perspectives on related dual quaternion matrices.