Eigenvectors and Eigenvalues of a Linear Map

Suppose I had a linear map such that and I want to find the eigenvalues and eigenvectors of T.

I'm doing this without determinants. I have it that

After rearranging terms and doing simple elimination I have

I believe the eigenvalue is then . Next to find the eigenvectors I would find

To find I would end up having that

and so I have . I believe this means that all eigenvectors would be in the form (i.e. (1,1),(2,2)...)

Was everything I did correct?

Thank you.