# Finding Rank from Characteristic Polynomial

• Apr 8th 2010, 04:21 PM
joe909
Finding Rank from Characteristic Polynomial
Let $A$ be a $6x6$ matrix with characteristic polynomial $x^3(x-1)(x+2)(x+4)$

A) Prove that the rank of $A$ is $3, 4$ or $5$.
B) If the rank of $A$ is 5 is $A$ diagonalizable?

A) I understand why rank must be less than $6$ because $A$ is clearly not signular since it has $0$ as an eigenvalue, however im having trouble actually proving that the rank must be $3, 4$ or $5$.

B) I think i may have this part. my thought is since Rank = $5$ Dim of the kernal must be $1$. Which means that the geometric multiplicity of the $0$ eigenvalue is $1$ where as its algebreic multiplicity is $3$ and since they are not equal it is not diagonalizable. If i understand right, it would be diagonalizable if $A$ was of rank $3$, correct?

Thanks for the help!