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
$$A=\begin{pmatrix}
0 &amp;0 &amp;0&amp; 1\\
1 &amp;0 &amp;0 &amp;0\\ 
0 &amp;1 &amp;0 &amp;0\\
0 &amp;0 &amp;1 &amp;0 
\end{pmatrix}$$

Then what is rank of $f(A)$? I think $f(A)$ will be a zero matrix so its rank is 0. Am I right?
Yes, that's exactly right...  $\phantom{ok I'll add a little random text here to avoid automatic conversion}{}{}$
From the definition of minimal polynomial $f(A)=0.$ Hence $f(A)$ is the zero transformation. Hence rank is $0.$