Question

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

2x1 + 3x2 − x3 = 14

x1 + 2x2 + x3 = 4

5x1 + 9x2 + 2x3 = 7

1 0 -5 16

0 1 3 -6

0 0 0 -19

Incorrect

Answer 1

The augmented matrix is

`\left[\begin{array}ccc2&3&-1&14\\1&2&1&4\\5&9&2&7\end{array}\right]`

Swap the first rows:

`\left[\begin{array}ccc1&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}ccc1&2&1&4\\0&-1&-3&6\\0&-1&-3&-13\end{array}\right]`

Multiply through row 2 by -1:

`\left[\begin{array}c1&2&1&4\\0&1&3&-6\\0&-1&-3&-13\end{array}\right]`

Add row 2 to row 3:

`\left[\begin{array}ccc1&2&1&4\\0&1&3&-6\\0&0&0&-19\end{array}\right]`

Multiply through row 3 by -1/19:

`\left[\begin{array}ccc1&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}ccc1&2&1&0\\0&1&3&0\\0&0&0&1\end{array}\right]`

Add -2(row 2) to row 1:

`\left[\begin{array}c1&0&-5&0\\0&1&3&0\\0&0&0&1\end{array}\right]`