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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.2012-12-08

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