The relationship between matrix rank and its characteristic polynomial coefficients

Question

Given the matrix characteristic polynomial coefficients. Is there a quick way to determine the rank of the matrix?

Answer 1

In general nothing can be said about rank of the matrix by merely looking at char polynomial. Take $$A=\left[\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}\right]$$ Rank$A=1$, though char polynomial is $x^2=0$. But if your matrix is diagonalizable, ''effective degree'' of the characteristic polynomial is equal to the matrix rank, since for a diagonalizable matrix, rank turns out to be number of non zero eigenvalues.

Answer 2

For a $n\times n$ matrix, if the characteristic polynomial is $X^mP(X)$ where $P$ is not divided by $X$, the rank is $n-m$.