Inverting and solving equations

Code:

1 -2 1
3 1 2
-1 4 1

Find the inverse of that matrix, hence solve the following

x-2y+z=1

I have 2 more equations to solve, but I just need a quick rundown of the process

To invert, I found the determinant (answer for it was

**-4** if anyone could confirm to see if I did the method right)

I then took the transpose, and prcoeeded to complete that big 3x3 thing which involves taking the multiplication of diagonal "\" and add it to the multiplication of diagonal "/".

The result of both the transpose and that process gave me this answer:

Code:

9 -6 5
-1 0 -5
11 -2 7

What made me doubt everything was that the 3 equations given simply don't match up with either the inverse of the provided matrix, with the matrix itself, or anything similar that I can recall.

Any help on this would be much appreciated.