Example NSR Nonsingular matrix, row-reduced

We have the coefficient matrix for Archetype B and a row-equivalent matrix $B$ in reduced row-echelon form, \begin{equation*} A=\begin{bmatrix} -7&-6&-12\\ 5&5&7\\ 1&0&4 \end{bmatrix} \rref\begin{bmatrix} \leading{1} & 0 & 0\\ 0 & \leading{1} & 0\\ 0 & 0 & \leading{1} \end{bmatrix} =B\end{equation*}

Since $B$ is the $3\times 3$ identity matrix, Theorem NMRRI tells us that $A$ is a nonsingular matrix.