Archetype A is the following system of 3 equations in 3 variables. \begin{align*} x_1 -x_2 +2x_3 & =1\\ 2x_1+ x_2 + x_3 & =8\\ x_1 + x_2 & =5 \end{align*} Here is its augmented matrix. \begin{align*} \begin{bmatrix} 1 & -1 & 2 & 1\\ 2 & 1 & 1 & 8\\ 1 & 1 & 0 & 5 \end{bmatrix} \end{align*}