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Let $\varrho :\mathbb{Z}\rightarrow GL_3(\mathbb{R})$ be the representation given by $\varrho (n)=A^n$ where

A=$\begin{pmatrix}

2 & 5 & -1 \\

2 & \frac{5}{2} & \frac{11}{2} \\

6 & \frac{-2}{2} & \frac{3}{2} \\

\end{pmatrix}$

Does ρ have any 1-dimensional invariant subspaces?

Do I have to divide up the elements in the representation with their corresponding eigenvectors. I know From $A^4$ the matrix is just a scaler of the previous 4, i.e. $A^4=I$, $A^5=A$ etc and their eigenvectors are the same then?

Im struggling to understand this chapter we are doing a little so any help is much appreciated.

Carla x

A=$\begin{pmatrix}

2 & 5 & -1 \\

2 & \frac{5}{2} & \frac{11}{2} \\

6 & \frac{-2}{2} & \frac{3}{2} \\

\end{pmatrix}$

Does ρ have any 1-dimensional invariant subspaces?

Do I have to divide up the elements in the representation with their corresponding eigenvectors. I know From $A^4$ the matrix is just a scaler of the previous 4, i.e. $A^4=I$, $A^5=A$ etc and their eigenvectors are the same then?

Im struggling to understand this chapter we are doing a little so any help is much appreciated.

Carla x

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