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I was wandering if there is a chance that two real matrices with the same characteristic polynomial have a different rank? I tried to prove it, but i failed. any suggestions?

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    Here's a related question: Can two real matrices with the same *minimal* polynomial have different rank?2011-09-18

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$\pmatrix{0 & 1\\ 0&0}$ and $\pmatrix{0&0\\0&0}$ both have characteristic polynomial $\lambda^2$.