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

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The discussion centers on determining the rank of the matrix expression \( B = A + A^2 + A^3 + A^4 \) given that the rank of matrix \( A \) is 2. It is established that the rank of \( B \) cannot be definitively determined from the rank of \( A \) alone. Specific examples illustrate that depending on the form of \( A \), the rank of \( B \) can vary: if \( A \) is the identity matrix, \( B \) has rank 2; if \( A \) is a diagonal matrix with values 1 and -1, \( B \) has rank 1; and if \( A \) is a diagonal matrix with -1, \( B \) has rank 0.

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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 $B=A+A^2+A^3+A^4$: $$\left \{ \begin{matrix}A=I\Rightarrow B=4I\Rightarrow\mbox{rank } B=2\\A=\begin{bmatrix}{1}&{\;0}\\{0}&{-1}\end{bmatrix}\Rightarrow B=\begin{bmatrix}{4}&{0}\\{0}&{0}\end{bmatrix} \Rightarrow\mbox{rank } B=1\\A=\begin{bmatrix}{-1}&{\;\;0}\\{\;\;0}&{-1}\end{bmatrix}\Rightarrow B=0\Rightarrow\mbox{rank } B=0\end{matrix}\right. $$
 

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