On semimonotone matrices of exact order two

In this paper, we introduce the notion of (strictly) semimonotone matrices of exact order $k$, where $0\leq k\leq n$, and explore their properties. We fully characterize the $3 \times 3$ (strictly) semimonotone matrices of exact order $2$, and show that the class of $3 \times 3$ semimonotone matrices of exact order $2$ forms a subclass of inverse $\mathbf{Z}$-matrices. We further investigate $n\times n$ (strictly) semimonotone matrices of exact order $2$, with emphasis on their identification and construction, and establish that every $n\times n$ semimonotone $\mathbf{Z}$-matrix of exact order $2$ is invertible. Additionally, we show that when $n-k=1$, the class of (strictly) semimonotone matrices of exact order $k$ is a subclass of $\mathbf{Z}$-matrices.