Archetype B is the system of $m=3$ linear equations in $n=3$ variables, \begin{align*} -7x_1 -6 x_2 - 12x_3 &=-33\\ 5x_1 + 5x_2 + 7x_3 &=24\\ x_1 +4x_3 &=5 \end{align*}
By Theorem SLEMM we can represent this system of equations as \begin{equation*} A\vect{x}=\vect{b} \end{equation*} where \begin{align*} A=\begin{bmatrix} -7&-6&-12\\ 5&5&7\\ 1&0&4 \end{bmatrix} &&\vect{x}=\colvector{x_1\\x_2\\x_3}&&\vect{b}=\colvector{-33\\24\\5} \end{align*}
Now, entirely unmotivated, we define the $3\times 3$ matrix $B$, \begin{equation*} B=\begin{bmatrix} -10 & -12 & -9\\ \frac{13}{2} & 8 & \frac{11}{2}\\ \frac{5}{2} & 3 & \frac{5}{2} \end{bmatrix}\end{equation*} and note the remarkable fact that \begin{equation*} BA=\begin{bmatrix} -10 & -12 & -9\\ \frac{13}{2} & 8 & \frac{11}{2}\\ \frac{5}{2} & 3 & \frac{5}{2} \end{bmatrix}\begin{bmatrix} -7&-6&-12\\ 5&5&7\\ 1&0&4 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} \end{equation*}
Now apply this computation to the problem of solving the system of equations, \begin{align*} \vect{x} &=I_3\vect{x}&&\knowl{./knowls/theorem.MMIM.knowl}{\text{Theorem MMIM}}\\ &=(BA)\vect{x}&&\text{Substitution}\\ &=B(A\vect{x})&&\knowl{./knowls/theorem.MMA.knowl}{\text{Theorem MMA}}\\ &=B\vect{b}&&\text{Substitution}\\ \end{align*}
So we have \begin{equation*} \vect{x}=B\vect{b}= \begin{bmatrix} -10 & -12 & -9\\ \frac{13}{2} & 8 & \frac{11}{2}\\ \frac{5}{2} & 3 & \frac{5}{2} \end{bmatrix}\colvector{-33\\24\\5}= \colvector{-3\\5\\2} \end{equation*}
So with the help and assistance of $B$ we have been able to determine a solution to the system represented by $A\vect{x}=\vect{b}$ through judicious use of matrix multiplication. We know by Theorem NMUS that since the coefficient matrix in this example is nonsingular, there would be a unique solution, no matter what the choice of $\vect{b}$. The derivation above amplifies this result, since we were forced to conclude that $\vect{x}=B\vect{b}$ and the solution could not be anything else. You should notice that this argument would hold for any particular choice of $\vect{b}$.