Dual orthogonal tripotent matrices

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