What is wrong with my matrix inversion?

[k_squared]

Homework Statement

Find the inverse of the matrix:

1 1 -1
2 -1 1
1 1 2

Homework Equations

One must be aware of the identity matrix, as well as how add one row to another with matrix multiplication, for example, the matrix

1 0 0
k 1 0
0 0 1

would add k times the first row to the second row.

The Attempt at a Solution

Which must be wrong, sadly enough.

 

[Fredrik]

After adding -5/2 times the third to the second (step 5), the number in the middle should be 1-5/2, but you wrote -3/5.

 

[Mark44]

Homework Statement

Find the inverse of the matrix:

1 1 -1
2 -1 1
1 1 2

Homework Equations

One must be aware of the identity matrix, as well as how add one row to another with matrix multiplication, for example, the matrix

1 0 0
k 1 0
0 0 1

would add k times the first row to the second row.

The Attempt at a Solution

Which must be wrong, sadly enough.

You're not doing it the way I learned to do this. How I learned was to set up an augmented matrix with the matrix to invert on the left, and the identity matrix on the right, like so:
$$\begin{bmatrix} 1 &1 & -1 &| & 1 & 0 & 0\\
2 & -1 & 1 & | & 0 & 1 & 0\\
1 & 1 & 2 & | & 0 & 0 & 1 \end{bmatrix}$$
Now completely row-reduce the matrix on the left to get it to the identity matrix, and your inverse will be on the right. When I did it, there were no "half" entries. Most of the entries had denominators of 3.

 

[JorisL]

First of is this the method you learned in class? Because this seems prone to errors.
You could try Gauss-Jordan Elimination it is essentially the same, but you have some "book keeping" abilities.

It is similar to Gaussian Elimination for systems of linear equations.

[Edit] Too late :)

 

[my2cts]

Just search for simple linear combinations.
Math is about using your brains.
If your method does not work invent a better one or study.

 

[SammyS]

Just search for simple linear combinations.
Math is about using your brains.
If your method does not work invent a better one or study.

It's true that OP would have benefited from using some simpler combinations.

However, the method works perfectly well. Fredrik has pointed out where a computational error occurred and Mark has suggest a method with easier "book keeping".