I am trying to figure out why the answer to the following question is (a) $M$ must be diagonalizable based on the given information.
I know that for $M$ to be diagonalizable the sum of the dimensions of the eigenspaces must equal $n$ — but we don't know what $n$ is. How can we determine what $n$ is, or is there some other way of knowing $M$ must be diagonalizable?
Suppose that $(t-2)^2(t-1)(t+2)t$ is the characteristic polynomial of a matrix $M$, and the null space of $M−2I$ has dimension $2$.
Which of (a), (b) or (c) below is the correct statement?
(a) $M$ must be diagonalizable.
(b) $M$ might be diagonalizable.
(c) $M$ cannot be diagonalizable.
Thanks in advance!