Eigenvectors and Eigenvalues of a Linear Map

Suppose I had a linear map $\displaystyle T:\mathbb{R}^2\rightarrow\mathbb{R}^2$ such that $\displaystyle T(x,y)=(10x+y,10y+x)$ and I want to find the eigenvalues and eigenvectors of T.

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

$\displaystyle 10x+y=\lambda{x}$

$\displaystyle 10y+x=\lambda{y}$

After rearranging terms and doing simple elimination I have

$\displaystyle -(10-\lambda)(10-\lambda)=0$

I believe the eigenvalue is then $\displaystyle \lambda=10$. Next to find the eigenvectors I would find $\displaystyle null(T-10I)$

To find $\displaystyle null(T-10I)$ I would end up having that

$\displaystyle 10x+y-10x=0$

$\displaystyle 10y+x-10y=0$

and so I have $\displaystyle x=y$. I believe this means that all eigenvectors would be in the form $\displaystyle (x,x)$ (i.e. (1,1),(2,2)...)

Was everything I did correct?

Thank you.