TITLE: Characteristic polynomial of a nilpotent matrix
QUESTION [11 upvotes]: Let $A$ be $n\times n$ nilpotent matrix.
How to calculate  its characteristic polynomial?
I know it should be $X^n$, but I don't know why?

REPLY [16 votes]: The minimal polynomial is of the form $x^k$ for some $k$, because the matrix is nilpotent. Since the minimal polynomial is divisible by all the irreducible polynomials which divide the characteristic polynomial, we see that in fact the only irreducible polynomial which divides the latter is $x$. Thus $\chi_A(x)=x^n$.