Is it okay to perform the same row operation twice on opposite rows?

Question

I am trying to find the inverse of the following matrix:

 1  2  3 
 2  1  4
 1  0  2

I draw the identity matrix next to it and start performing row operations. After a few I get this matrix:

 1  2  3 |  1  0  0
 0 -3 -2 | -2  1  0
 0 -2 -1 | -1  0  1

I want to get a one in the middle of the middle row so I multiply the bottom row by -2 and add it to the middle row to get the following:

 1  2  3 |  1  0  0
 0  1  0 |  0  1 -2
 0 -2 -1 | -1  0  1

Then in order to get rid of the first -2 in the bottom row I perform exactly the reverse operation (multiplying the middle row by -2 and adding it to the bottom row) to get the following:

 1  2  3 |  1  0  0
 0  1  0 |  0  1 -2
 0  0 -1 | -1 -2  5

However, comparing this to the final matrix shown below, I will never be able to adjust the bottom row to compensate. Was the operation I performed allowed or did I simply make a mistake?

 1  0  0 | -2  4 -5
 0  1  0 |  0  1 -2
 0  0  1 |  1 -2  3

Answer 1

I simply made a mistake in row operations. I had:

1  2  3 |  1  0  0
0  1  0 |  0  1 -2
0 -2 -1 | -1  0  1

In the next step, multiplying the middle row by 2 and adding to the bottom row, should give me

1  2  3 |  1  0  0
0  1  0 |  0  1 -2
0  0 -1 | -1  2 -3

Which I can then use to find the inverse (the correct solution is given in the question).