Hello, I've been looking at question which states to diagonalize the matrix

I've found the eigenvalues to be 1, 0 and t. I know that if t = 1, the matrix is not diagonalizable, and if t = 0, then A is already diagonalized, and thus is diagonalizable. Now, I have to look at what happens when t is not equal to 1 or 0.

Looking at each eigenvalue, I'm having trouble finding the eigenvectors.

For , they say, "Obviously , therefore is a eigenvector." I don't follow that. It's not obvious to me.

For , I know that this is the same as just finding Ax = 0, however, my eigenvector is , & the one the chose was . I can't see where I went wrong.

For , they have the eigenvector being I'm completely lost there.

How are all these eigenvectors found?

Thanks for your time.