# Finding Rank from Characteristic Polynomial

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

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

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

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

Thanks for the help!