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Let $A,B\in M_{n}(C)$ such that $A$ satisfies in the characteristic polynomial of $B$ and $B$ satisfies in the characteristic polynomial of $A$. Is the the characteristic polynomial of $A$ is equal to the characteristic polynomial of $B$? Is $rank(A)=rank(B)$? Is $A$ diagonalize if and only if $B$ diagonalize?

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Don't forget to accept the answer which you like. It makes to receive answers later. –  B. S. Dec 8 '12 at 9:29
    
Why don't you accept @Euyu's answer? –  Did Jan 1 '13 at 19:07
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1 Answer 1

None of what you mentioned needs to be true. The matrices do need to have the same eigenvalues (not up to multiplicity) though since the minimal polynomial of each divides the characteristic polynomial of the other. In particular, this means that $A$ will be invertible if and only if $B$ is invertible.

But in general, you can have matrices which satisfy each other's characteristic polynomials, but have different rank, different diagonalizability and different characteristic polynomials.

Let $A$ be a matrix with minimal polynomial $x(x-1)$ and characteristic polynomial $x^{n-1}(x-1)$.

Let $B$ be a matrix with minimal polynomial $x^2(x-1)$ and characteristic polynomial $x^2(x-1)^{n-2}$.

The two matrices each satisfy the other's characteristic polynomial. $A$ is diagonalizable but $B$ is not. $A$ has rank $1$ while $B$ has rank $n-1$.

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