On factorizations of upper triangular nonnegative matrices of order three (Q1723589)

Summary: Let \(T_3(\mathbb{N}_0)\) denote the semigroup of \(3 \times 3\) upper triangular matrices with nonnegative integral-valued entries. In this paper, we investigate factorizations of upper triangular nonnegative matrices of order three. Firstly, we characterize the atoms of the subsemigroup \(S\) of the matrices in \(T_3(\mathbb{N}_0)\) with nonzero determinant and give some formulas. As a consequence, problems 4a and 4c presented by \textit{N. Baeth} et al. [Linear Algebra Appl. 434, No. 3, 694--711 (2011; Zbl 1250.11104)] are each half-answered for the case \(n = 3\). And then, we consider some factorization cases of matrix \(A\) in \(S\) with \(\rho(A) = 1\) and give formulas for the minimum factorization length of some special matrices in \(S\).