Answer
final matrix:
$\begin{bmatrix} \underline{1}&0&-1&0\\ 0&\underline{1}&2&0\\ 0&0&0&\underline{1} \end{bmatrix}$
original matrix:
$\begin{bmatrix} \underline{1}&3&5&7\\ 3&\underline{5}&7&9\\ 5&7&9&\underline{1} \end{bmatrix}$
The underlined positions are the pivot positions.
The pivot columns are coumns 1, 2 and 4.
Work Step by Step
$\begin{bmatrix} 1&3&5&7\\ 3&5&7&9\\ 5&7&9&1 \end{bmatrix}$
Row 2 = Row 2 - 3*Row 1
$\begin{bmatrix} 1&3&5&7\\ 0&-4&-8&-12\\ 5&7&9&1 \end{bmatrix}$
Row 2 = Row 2 / -4
$\begin{bmatrix} 1&3&5&7\\ 0&1&2&3\\ 5&7&9&1 \end{bmatrix}$
Row 3 = Row 3 - 7*Row 2
$\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 5&0&-5&-20 \end{bmatrix}$
Row 3 = Row 3 / 5
$\begin{bmatrix} 1&3&5&7\\ 0&1&2&3\\ 1&0&-1&-4 \end{bmatrix}$
Row 1 = Row 1 - 3*Row 2
$\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 1&0&-1&-4 \end{bmatrix}$
Row 3 = Row 3 - Row 1
$\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 0&0&0&-2 \end{bmatrix}$
Row 3 = Row 3 / -2
$\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&3\\ 0&0&0&1 \end{bmatrix}$
Row 2 = Row 2 - 3*Row 3
$\begin{bmatrix} 1&0&-1&-2\\ 0&1&2&0\\ 0&0&0&1 \end{bmatrix}$
