I believe you have to just work out the eigenvectors and check. I'll double check this.
The number of times an eigenvalue is repeated is called it's algebraic multiplicity.
The number of independent eigenvectors for a repeated eigenvalue is called it's geometric multiplicity.
The geometric multiplicity <= the algebraic multiplicity
If it is less than, then the original matrix is called "defective"
I'm not seeing any clever test on a matrix other than checking it's eigenvectors for linear independence to determine if it is defective.
it looks like you correctly came up with $(0,0,1)$ and $(1,1,0)$ for the two eigenvectors associated with the eigenvalue 3.
These two vectors are pretty clearly linearly independent.
So those two along with the eigenvector associated with the eigenvalue 1 make up a set of 3 linearly independent eigenvectors.