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Problem of Rank of a matrix

suvadip

Member
Feb 21, 2013
69
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

Well-known member
MHB Math Helper
Jan 29, 2012
661
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 $B=A+A^2+A^3+A^4$: $$\left \{ \begin{matrix}A=I\Rightarrow B=4I\Rightarrow\mbox{rank } B=2\\A=\begin{bmatrix}{1}&{\;0}\\{0}&{-1}\end{bmatrix}\Rightarrow B=\begin{bmatrix}{4}&{0}\\{0}&{0}\end{bmatrix} \Rightarrow\mbox{rank } B=1\\A=\begin{bmatrix}{-1}&{\;\;0}\\{\;\;0}&{-1}\end{bmatrix}\Rightarrow B=0\Rightarrow\mbox{rank } B=0\end{matrix}\right. $$