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
