Example RRTI Rank, rank of transpose, Archetype I

Archetype I has a $4\times 7$ coefficient matrix which row-reduces to \begin{equation*} \begin{bmatrix} \leading{1} & 4 & 0 & 0 & 2 & 1 & -3\\ 0 & 0 & \leading{1} & 0 & 1 & -3 & 5\\ 0 & 0 & 0 & \leading{1} & 2 & -6 & 6\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{equation*} so the rank is $3$. Row-reducing the transpose yields \begin{equation*} \begin{bmatrix} \leading{1} & 0 & 0 & -\frac{31}{7}\\ 0 & \leading{1} & 0 & \frac{12}{7}\\ 0 & 0 & \leading{1} & \frac{13}{7}\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}. \end{equation*} demonstrating that the rank of the transpose is also $3$.