On semimonotone matrices of exact order two

In this paper, we introduce the notion of (strictly) semimonotone matrices of exact order k, where 0≤𝑘≤𝑛, and explore their properties. We fully characterize the 3×3 (strictly) semimonotone matrices of exact order 2, and show that the class of 3×3 semimonotone matrices of exact order 2 forms a subclass of inverse 𝐙-matrices. We further investigate 𝑛×𝑛 (strictly) semimonotone matrices of exact order 2, with emphasis on their identification and construction, and establish that every 𝑛×𝑛 semimonotone 𝐙-matrices 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 𝐙-matrices.