We have the coefficient matrix for Archetype A and a row-equivalent matrix $B$ in reduced row-echelon form, \begin{equation*} A=\begin{bmatrix} 1 & -1 & 2\\ 2 & 1 & 1\\ 1 & 1 & 0 \end{bmatrix} \rref\begin{bmatrix} \leading{1} & 0 & 1\\ 0 & \leading{1} & -1\\ 0 & 0 & 0 \end{bmatrix} =B\end{equation*}
Since $B$ is not the $3\times 3$ identity matrix, Theorem NMRRI tells us that $A$ is a singular matrix.