I'm stock on finding the Inverse of a 3 x 3 Matrix - My head hurts - Please HELP!

The given matrix is:

3 5 2
4 7 7
5 8 9

So far here's what I got:

1........ 1.67...... 0.67 | 0.03.....0..........0
0.62.......1........ ..0 | 0.33....-0.095...0
0.03.......0.......... 1 | -0.28....0.........0.172

And so I'm really confused now on how to get the rest of the zeros here... could you please help me though this?

Thank you so much!

You would subtract 0.62 times the first row from the second row, which would get rid of the leading 0.62. You would subtract 0.03 times the first row from the third row, which would get rid of the leading 0.03. Similar operations would get rid of the 1.67 and 0.67. You would have normalize to all rows at the end to put 1s along the diagonal.

But in general, you're sort of doing it (i.e. Gaussian elimination) wrong. You shouldn't make things into 1's so early if you can avoid it, and you should stick with fractions where possible. Also remember that you can multiply a row through by a constant to use whole numbers rather than fractions. For instance, as a first step, I would replace the third row with 4 times the third row minus 5 times the second row, i.e.

4*(5, 8, 9, 0, 0, 1) - 5*(4, 7, 7, 0, 1, 0) = (20-20, 32-35, 36-35, 0, 0-5, 4-0) = (0, -3, 1, 0, -5, 4), giving
3 5 2 | 1 0 0
4 7 7 | 0 1 0
0 -3 1 | 0 -5 4

I would then do basically the same thing to the second row, giving
3 5 2 | 1 0 0
0 1 13 | -4 3 0
0 -3 1 | 0 -5 4

Now add three times the second row to the third row:
3 5 2 | 1 0 0
0 1 13 | -4 3 0
0 0 40 | -12 4 4

Divide the fourth row by 4 to make it simpler:
3 5 2 | 1 0 0
0 1 13 | -4 3 0
0 0 10 | -3 1 1

Subtract off 5 times the second row from the first row:
3 0 -63 | 21 -15 0
0 1 13 | -4 3 0
0 0 10 | -3 1 1

Divide the first row by 3 to make it simpler:
1 0 -21 | 7 -5 0