## Linear Algebra and Its Applications (5th Edition)

$\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 0&0&0&0 \end{bmatrix}$ The pivots are the the entries in positions $(1, 1)$ and $(2, 2)$ in both the original and the final matrices, where coordinates are in $(row, column)$ form. The pivot columns are therefore columns one and two.
(1) Begin with the original matrix. $\begin{bmatrix} 1&2&3&4\\ 4&5&6&7\\ 6&7&8&9 \end{bmatrix}$ (2) Add $-4$ times row one to row two. Add $-6$ times row one to row three. $\begin{bmatrix} 1&2&3&4\\ 0&-3&-6&-9\\ 0&-5&-10&-15 \end{bmatrix}$ (3) Multiply row two by $-\frac{1}{3}$. Multiply row three by $-\frac{1}{5}$. $\begin{bmatrix} 1&2&3&4\\ 0&1&2&3\\ 0&1&2&3 \end{bmatrix}$ (4) Add $-1$ times row two to row three. Add $-2$ times row two to row one. $\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 0&0&0&0 \end{bmatrix}$