It hard to understand what you are saying because you do not give the original matrix but I think you have a matrix which has an eienvalue of -2 and,afteryou subtract -2 from each diagonal element, you have the matrix

.

Now, you are seeking an eigenvector, a matrix, such that

To do that you can row reduce the matrix (you don't need to list the final column of the augmented matrix since that is always 0).

But you seem to be doing the row operations pretty much at random. I think you will find it simpler to always work on one column at a time, from the left, getting a "1" at the pivot and "0"s elsewhere. To reduce the first column, divide the row by 7, then add 2 times that new first row to the second row, then subract twice the new first row from the third. That gives

Now, it is easy to see that we can divide the second row by 9/7, subrtact 8/7 times that new second row from the first, and subtract 12/7 times that new second row from the first to get

Now that, remember, is the same as saying

That tells us that we must have x- z= 0 or z= x, and y= 0. That is, an eigenvector is of the form .