# Given a characteristic polynomial of a matrix A, find the nullity, rank

If i'm only given the characteristic polynomial of a matrix $$A$$, how would I find the given information:

• Rank
• Nullity
• Number of elements in the eigenvectors of $$A$$
• Number of elements in the null space of $$A$$

The only information that i can figure out are:

• the eigenvalues
• the dimensions of the matrix and
• the determinant of the matrix which is the product of the eigenvalues.

If you only have the characteristic polynomial, then you can’t find much about the rank or nullity of $$A$$. Consider the following examples:
$$A_1=\begin{bmatrix} 0 & 1\\0&0\end{bmatrix};\qquad A_2 = \begin{bmatrix}0&0\\0&0\end{bmatrix}.$$
The characteristic polynomial of both matrices is $$x^2$$, but the rank and nullity of $$A_1$$ is different from that of $$A_2$$.
• The most information you can gather about rank or nullity from the characteristic polynomial is that nullity is at least $1$ if $x$ is a factor, and the rank is at least $k$ if there are $k$ distinct linear factors not equal to $x$. – Santana Afton Mar 9 at 18:39