# If the characteristic polynomial of matrix $A$ has $n$ zero roots, then $A$ is nilpotent.

How can I prove the following statement:

If the characteristic polynomial of matrix $A$ has $n$ zero roots, then $A$ is nilpotent.

Thank you!

• Is $A$ an $n \times n$ matrix? Jul 30 '13 at 13:24
• See the answers to this question.
– jkn
Jul 30 '13 at 13:25
• Have you heard of the Cayley-Hamilton theorem? Jul 30 '13 at 13:26

If the characteristic polynomial has only zero roots, it has the form $\lambda^n = 0$ (or $(-\lambda)^n = 0$, depending on convention). By the theorem of Cayley-Hamilton, a square matrix $A$ fulfills its own characteristic equation (even if it's not diagonalizable), therefore $A^n = 0$ (zero matrix), therefore $A$ is nilpotent.