Question

For the following linear system, put the augmented coefficient matrix into reduced row-echelon form.

Answer 1

The augmented matrix is

$\left[\begin{array}{ccc|c}2&3&-1&14\\1&2&1&4\\5&9&2&7\end{array}\right]$

Swap the first rows:

$\left[\begin{array}{ccc|c}1&2&1&4\\2&3&-1&14\\5&9&2&7\end{array}\right]$

Add -2(row 1) to row 2, and add -5(row 1) to row 3:

$\left[\begin{array}{ccc|c}1&2&1&4\\0&-1&-3&6\\0&-1&-3&-13\end{array}\right]$

Multiply through row 2 by -1:

$\left[\begin{array}{ccc|c}1&2&1&4\\0&1&3&-6\\0&-1&-3&-13\end{array}\right]$

Add row 2 to row 3:

$\left[\begin{array}{ccc|c}1&2&1&4\\0&1&3&-6\\0&0&0&-19\end{array}\right]$

Multiply through row 3 by -1/19:

$\left[\begin{array}{ccc|c}1&2&1&4\\0&1&3&-6\\0&0&0&1\end{array}\right]$

Add -4(row 3) to row 1, and add 6(row 3) to row 2:

$\left[\begin{array}{ccc|c}1&2&1&0\\0&1&3&0\\0&0&0&1\end{array}\right]$

Add -2(row 2) to row 1:

$\left[\begin{array}{ccc|c}1&0&-5&0\\0&1&3&0\\0&0&0&1\end{array}\right]$