I need help with this problem I came across. I give it in a simplified way:

The questions asks for the maximum number of linearly independent eigenvectors for the eigenvalue 7.

Now suppose the Matrix we are dealing with is in 3D and has eigenvectors:

, the k's are natural numbers (including zero).

Now the corresponding eigenvalues are:

.

\

\

What I thought at first was that since the expression has eigenvalue 3, adding one more term would mean adding something of the form , but the added expression would have eigenvalue 8. So at most we can have 1 eigenvector.

\

\

I'm not sure if just adding eigenvalues in the way I did is allowed.

UPDATE:

I was thinking maybe the question meant the following:

Given the eigenvalue , if we equate this to seven, then we obtain , in which case there are a maximum of 6 linearly independent eigevectors:

, ,

, ,

Which is Right?

-Thanks!