# Is it true that two real matrices with the same characteristic polynomial have the same rank?

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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Take the zero two by two matrix and a nonzero zero two by two nilpotent matrix. –  Pierre-Yves Gaillard Sep 18 '11 at 15:51
Here's a related question: Can two real matrices with the same minimal polynomial have different rank? –  alex.jordan Sep 18 '11 at 18:50

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