MHB Can Rank of A Determine Rank of A+A²+A³+A⁴?

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The rank of the matrix expression A + A² + A³ + A⁴ cannot be determined solely from the rank of A being 2. Various examples illustrate that depending on the specific form of matrix A, the rank of the resulting matrix B can vary significantly. For instance, if A is the identity matrix, B has a rank of 2, while for other configurations, B can have a rank of 1 or even 0. Therefore, the relationship between the rank of A and the rank of B is not straightforward. The conclusion is that additional information about matrix A is necessary to ascertain the rank of B.
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If rank of A is 2. Is it possible to find the rank of A+A2+A3+A4

from that information? Please help
 
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suvadip said:
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 $B=A+A^2+A^3+A^4$: $$\left \{ \begin{matrix}A=I\Rightarrow B=4I\Rightarrow\mbox{rank } B=2\\A=\begin{bmatrix}{1}&{\;0}\\{0}&{-1}\end{bmatrix}\Rightarrow B=\begin{bmatrix}{4}&{0}\\{0}&{0}\end{bmatrix} \Rightarrow\mbox{rank } B=1\\A=\begin{bmatrix}{-1}&{\;\;0}\\{\;\;0}&{-1}\end{bmatrix}\Rightarrow B=0\Rightarrow\mbox{rank } B=0\end{matrix}\right. $$
 
Thread 'How to define a vector field?'

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