Possible dimensions of eigenspaces from characteristic equation (x²)(x-1)(x-2)³

[georgeh]

So the question states:
Let A be a 6x6 matrix with characteristic equation (x^2)(x-1)(x-2)^3=0
What are the possible dimensions of eigenspaces of A?
so..
eigen values possible are
x=1,x=0,x=2
for x = 0
dimension 0?
for x = 1
I would say at dimension 0 or 1

x = 2
Dimension 0, 1, 2, 3
They have for x= 0
1 or 2
for x = 1
1 dimension
for x =2: 1,2,or 3
I am not sure how they got these answers. Any help would be appreciated.

 

[JFo]

for a given eigenvalue $\lambda$, the dimension of the corresponding eigenspace $E_{\lambda}$ must satisfy

$$1 \leq dim(E_{\lambda}) \leq multiplicity(\lambda)$$