As a vector equality, Archetype A can be written as \begin{equation*} \colvector{x_1 -x_2 +2x_3\\2x_1+ x_2 + x_3\\ x_1 + x_2} = \colvector{1\\8\\5} \end{equation*}
Now bust up the linear expressions on the left, first using vector addition, \begin{equation*} \colvector{x_1\\2x_1\\x_1}+ \colvector{-x_2\\x_2\\x_2}+ \colvector{2x_3\\x_3\\0x_3} = \colvector{1\\8\\5} \end{equation*}
Rewrite each of these vectors as a scalar multiple of a fixed vector, where the scalar is one of the unknown variables, converting the left-hand side into a linear combination \begin{equation*} x_1\colvector{1\\2\\1}+ x_2\colvector{-1\\1\\1}+ x_3\colvector{2\\1\\0} = \colvector{1\\8\\5} \end{equation*}
Row-reducing the augmented matrix for Archetype A leads to the conclusion that the system is consistent and has free variables, hence infinitely many solutions. So for example, the two solutions \begin{align*} x_1 = 2&&x_2 = 3&&x_3 = 1\\ x_1 = 3&&x_2 = 2&&x_3 = 0 \end{align*} can be used together to say that, \begin{equation*} (2)\colvector{1\\2\\1}+ (3)\colvector{-1\\1\\1}+ (1)\colvector{2\\1\\0} = \colvector{1\\8\\5} = (3)\colvector{1\\2\\1}+ (2)\colvector{-1\\1\\1}+ (0)\colvector{2\\1\\0} \end{equation*} Ignore the middle of this equation, and move all the terms to the left-hand side, \begin{equation*} (2)\colvector{1\\2\\1}+ (3)\colvector{-1\\1\\1}+ (1)\colvector{2\\1\\0}+ (-3)\colvector{1\\2\\1}+ (-2)\colvector{-1\\1\\1}+ (-0)\colvector{2\\1\\0} = \colvector{0\\0\\0} \end{equation*} Regrouping gives \begin{equation*} (-1)\colvector{1\\2\\1}+ (1)\colvector{-1\\1\\1}+ (1)\colvector{2\\1\\0} = \colvector{0\\0\\0} \end{equation*}
Notice that these three vectors are the columns of the coefficient matrix for the system of equations in Archetype A. This equality says there is a linear combination of those columns that equals the vector of all zeros. Give it some thought, but this says that \begin{align*} x_1=-1&&x_2=1&&x_3=1 \end{align*} is a nontrivial solution to the homogeneous system of equations with the coefficient matrix for the original system in Archetype A. In particular, this demonstrates that this coefficient matrix is singular.