Archetype B has a $3\times 3$ coefficient matrix \begin{equation*} B=\begin{bmatrix} -7&-6&-12\\ 5&5&7\\ 1&0&4 \end{bmatrix} \end{equation*} and is similar to a diagonal matrix, as can be seen by the following computation with the nonsingular matrix $S$, \begin{align*} \similar{B}{S}&= \inverse{\begin{bmatrix}-5&-3&-2\\3&2&1\\1&1&1\end{bmatrix}}\begin{bmatrix} -7&-6&-12\\ 5&5&7\\ 1&0&4 \end{bmatrix} \begin{bmatrix}-5&-3&-2\\3&2&1\\1&1&1\end{bmatrix}\\ &=\begin{bmatrix}-1&-1&-1\\2&3&1\\-1&-2&1\end{bmatrix}\begin{bmatrix} -7&-6&-12\\ 5&5&7\\ 1&0&4 \end{bmatrix} \begin{bmatrix}-5&-3&-2\\3&2&1\\1&1&1\end{bmatrix}\\ &= \begin{bmatrix}-1&0&0\\0&1&0\\0&0&2\end{bmatrix} \end{align*}