Omitting an 'optional' step in gaussian-elimination causes different result

Question

I'm trying to invert a matrix using the gaussian-elimination method

This is my start matrix

+---------+--------+
| 1 -2 -2 | 1 0 0  |
| -2 1 -2 | 0 1 0  |
| -2 -2 1 | 0 0 1  |
+---------+--------+

1. Using the top left 1 as pivot I can eliminate the first column, the result is

+---------+--------+
| 1 -2 -2 | 1 0 0  |
| 0 -3 -6 | 2 1 0  |
| 0 -6 -3 | 2 0 1  |
+---------+--------+

2. Selecting the middle -3 as pivot, I eliminate the second column, the result is

+---------+--------+
| -3 0 -6 |  1 2 0 |
| 0 -3 -6 | 2 1 0  |
| 0 0 -9  | 2 2 -1 |
+---------+--------+

3. And finally selecting the bottom right -9as pivot, I eliminate the last column, the result is

+--------+----------+
| 9 0 0  |  1 -2 -2 |
| 0 -9 0 | -2 1 -2  |
| 0 0 -9 | 2 2 -1   |
+--------+----------+

4. Simplifying gives

+-------+---------------+
| 1 0 0 | 1/9 -2/9 -2/9 |
| 0 1 0 | 2/9 -1/9 2/9  |
| 0 0 1 | -2/9 -2/9 1/9 |
+-------+---------------+

But when checking the answer online, I found out my result is wrong, it should be

+-------+---------------+
| 1 0 0 | 1/9 -2/9 -2/9 |
| 0 1 0 | -2/9 1/9 -2/9 |
| 0 0 1 | -2/9 -2/9 1/9 |
+-------+---------------+

(difference on the middle row, the numbers on the second row on the right side should be multiplied by -1)

I checked on different sites and they all showed me to divide row 2 by -3 after step 1.

For example on another site

What I do not understand is why they divide the second row by -3. Isn't this an optional step? Shouldn't it make no difference when not dividing by -3 or are multiple answers possible?

Answer 1

After step 3 before step 4, the result should be

$$\begin{align*} \pmatrix{-3&0&-6&1&2&0\\ 0&-3&-6&2&1&0\\ 0&0&-9&2&2&-1} &\xrightarrow{R_1\to -3R_1+2R_3} \pmatrix{9&0&0&1&-2&-2\\ 0&-3&-6&2&1&0\\ 0&0&-9&2&2&-1}\\ &\xrightarrow{R_2\to3R_2-2R_3} \pmatrix{9&0&0&1&-2&-2\\ 0&-9&0&2&-1&2\\ 0&0&-9&2&2&-1} \end{align*}$$