Polynomials which permute matrices over commutative antinegative semirings (Q1184483)

On any commutative, antinegative semiring \(\mathbb{S}\) (e.g. \(\mathbb{R}_ +,\mathbb{Z}_ +)\) the operator \(X\mapsto\sum^ m_{k=0}a_ kX^ k\), \(a_ k\in\mathbb{S}\), \(X\in\mathbb{S}^{n\times n}\), \(n>1\) is surjective iff \(a_ 1\) is a unit and all other \(a_ k=0\). The proof is carried out for \(n=2\).