How to determine if something is an eigenvalue of A without knowing A

Question

I'm having trouble with a true/false question on my Linear Algebra homework...

Let A be an n x n matrix. If dim Nul(A-2 I) = 3 then λ = 2 is an eigenvalue of A.

I know that dim Nul(A-2 I) = 3 tells me that the dimension of the eigenspace is 3, so the span contains 3 vectors; however, I'm not sure how this connects to λ = 2 being an eigenvalue...I would assume that it is, but I don't know how to prove it.

Answer 1

It's the definition of eigenvalue. $\lambda$ is an eigenvalue if $A-\lambda I$ is not invertible. And a matrix is invertible if and only if it has a nontrivial nullspace. If you know that $A-2I$ has a nullspace of dimension 3, then it is not invertible, hence 2 is an eigenvalue.