Archetype A can be converted to the homogeneous system, \begin{align*} x_1 -x_2 +2x_3 & = 0\\ 2x_1+ x_2 + x_3 & = 0\\ x_1 + x_2\quad\quad & = 0 \end{align*} whose augmented matrix row-reduces to \begin{equation*} \begin{bmatrix} \leading{1} & 0 & 1 & 0 \\ 0 & \leading{1} & -1 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix} \end{equation*}
By Theorem HSC, the system is consistent, and so the computation $n-r=3-2=1$ means the solution set contains one free variable by Theorem FVCS, and hence has infinitely many solutions. We can describe this solution set using the free variable $x_3$, \begin{equation*} S=\setparts{\colvector{x_1\\x_2\\x_3}}{x_1=-x_3,\,x_2=x_3} =\setparts{\colvector{-x_3\\x_3\\x_3}}{x_3\in\complex{\null}} \end{equation*}
Geometrically, these are points in three dimensions that lie on a line through the origin.