Is there some kind of relation between the rank of the matrix and its characteristic polynomial?After searching through various posts
Say if $A \in M_{5}(\Bbb{R})$ and its characteristic polynomial is $\alpha x^5 + \beta x^4 + \gamma x^3 =0 $,then the rank of matrix $A$ = ?
I am unable to estalish the relation ,like I know that from characteristic polynomial i can obtain the eigenvalues and hence the trace and determinant of the matrix and now the question is if i know the trace and determinat of the matrix can i obtain some information about the rank of the matrix(the number of linearly independent rows in the rref).
I was looking at this question but still i am not aware of any trick or relation.
Best Answer
Let $A$ be an $n$-by-$n$ matrix, and suppose that $0$ is a zero of the characteristic polynomial with multiplicity $m\ge1$. Then the rank of $A$ can be any number between $n-m$ and $n-1$ inclusive.