Example S A singular matrix, Archetype A

Example HISAA shows that the coefficient matrix derived from Archetype A, specifically the $3\times 3$ matrix, \begin{equation*} A=\begin{bmatrix} 1 & -1 & 2\\ 2 & 1 & 1\\ 1 & 1 & 0 \end{bmatrix} \end{equation*} is a singular matrix since there are nontrivial solutions to the homogeneous system $\homosystem{A}$.