# What is rank of $f(A)$, where $f$ is the minimal polynomial of $A$?

If $f(x)$ is minimal polynomial of the $4\times 4$ matrix $$A=\begin{pmatrix} 0 &0 &0& 1\\ 1 &0 &0 &0\\ 0 &1 &0 &0\\ 0 &0 &1 &0 \end{pmatrix}$$

Then what is rank of $f(A)$? I think $f(A)$ will be a zero matrix so its rank is 0. Am I right?

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Yes, that's exactly right. – Robert Israel Dec 24 '12 at 3:57

Yes, that's exactly right... $\phantom{ok I'll add a little random text here to avoid automatic conversion}{}{}$