Find the inverse of

[1 -1  0]
[0  1  1]
[1  0 -1]

Put the identity matrix on the right
(The identity matrix is a 3x3 matrix
which has 1's on the main diagonal and
0's elsewhere):

[1 -1  0 | 1  0  0]
[0  1  1 | 0  1  0]
[1  0 -1 | 0  0  1]

Make the identity matrix on the left.

Get a 0 where the first 1 is in the bottom row:
multiply the 1st row by -1, add it to 1 times
the 3rd row and restore the 1st row:

-1[1 -1  0 | 1  0  0]
  [0  1  1 | 0  1  0]
 1[1  0 -1 | 0  0  1]

  [1 -1  0 | 1  0  0]
  [0  1  1 | 0  1  0]
  [0  1 -1 |-1  0  1]

Get a 0 where the -1 is in the top row:
multiply the 2nd row by 1, add it to 1 times
the 1st row and restore the 2nd row: 

 1[1 -1  0 | 1  0  0]
 1[0  1  1 | 0  1  0]
  [0  1 -1 |-1  0  1]

  [1  0  1 | 1  1  0]
  [0  1  1 | 0  1  0]
  [0  1 -1 |-1  0  1]

Get a 0 where the first 1 is in the bottom row:
multiply the 2nd row by -1, add it to 1 times
the 3rd row and restore the 2nd row:

  [1  0  1 | 1  1  0]
-1[0  1  1 | 0  1  0]
 1[0  1 -1 |-1  0  1]

  [1  0  1 | 1  1  0]
  [0  1  1 | 0  1  0]
  [0  0 -2 |-1 -1  1]

Get a 0 where the second 1 is in the 1st row:
multiply the 3rd row by 1, add it to 2 times
the 1st row and restore the 3rd row:

2[1  0  1 | 1  1  0]
 [0  1  1 | 0  1  0]
1[0  0 -2 |-1 -1  1]

 [2  0  0 | 1  1  1]
 [0  1  1 | 0  1  0]
 [0  0 -2 |-1 -1  1]

Get a 0 where the second 1 is in the 2nd row:
multiply the 3rd row by 1, add it to 2 times
the 2nd row and restore the 3rd row:

 [2  0  0 | 1  1  1]
2[0  1  1 | 0  1  0]
1[0  0 -2 |-1 -1  1]

 [2  0  0 | 1  1  1]
 [0  2  0 |-1  1  1]
 [0  0 -2 |-1 -1  1]

To get a 1 where the 2 is in the 1st row,
multiply the 1st row by .5

.5[2  0  0 | 1  1  1]
  [0  2  0 |-1  1  1]
  [0  0 -2 |-1 -1  1]

  [1  0  0 |.5 .5 .5]
  [0  2  0 |-1  1  1]
  [0  0 -2 |-1 -1  1]

To get a 1 where the 2 is in the 2nd row,
multiply the 2nd row by .5

  [1  0  0 |.5 .5 .5]
.5[0  2  0 |-1  1  1]
  [0  0 -2 |-1 -1  1]

  [1  0  0 | .5 .5 .5]
  [0  1  0 |-.5 .5 .5]
  [0  0 -2 | -1 -1  1]

To get a 1 where the -2 is in the 3rd row,
multiply the 2nd row by -.5

  1[1  0  0 | .5 .5 .5]
   [0  1  0 |-.5 .5 .5]
-.5[0  0 -2 | -1 -1  1]

  [1  0  0 | .5 .5  .5]
  [0  1  0 |-.5 .5  .5]
  [0  0  1 | .5 .5 -.5]

Now that we have the identity on the left,
the matrix on the right is the inverse:

           [ .5 .5  .5]
           [-.5 .5  .5]
           [ .5 .5 -.5]

Edwin