So I've been doing this for a VERY long time now I don't seem to be getting the right answer (according to online solvers) I'm wondering if you guys can spot my error because this is REALLY frustrating.

And I found one of the Eigenvalues to be 1. so now subtracting the Diagonal from 1 I get this and I start to solve the homogeneous system of equation.

=

=

Now from here x(2)=T and x(1)+(1/3)T=0 --->X(1)=(-1/3)T

SO

BUT according toe the online solver it should be

I have tried everything so I need someone's help, WHAT AM I DOING WRONG?

Thank you!