Problem of Rank of a matrix

[suvadip]

If rank of A is 2. Is it possible to find the rank of A+A²+A³+A⁴

from that information? Please help

 

[Fernando Revilla]

If rank of A is 2. Is it possible to find the rank of A+A²+A³+A⁴

from that information? Please help

It can't be determined, denoting $B=A+A^2+A^3+A^4$: $$\left \{ \begin{matrix}A=I\Rightarrow B=4I\Rightarrow\mbox{rank } B=2\\A=\begin{bmatrix}{1}&{\;0}\\{0}&{-1}\end{bmatrix}\Rightarrow B=\begin{bmatrix}{4}&{0}\\{0}&{0}\end{bmatrix} \Rightarrow\mbox{rank } B=1\\A=\begin{bmatrix}{-1}&{\;\;0}\\{\;\;0}&{-1}\end{bmatrix}\Rightarrow B=0\Rightarrow\mbox{rank } B=0\end{matrix}\right. $$