I am asked to find the upper element of the vector $A^n = \begin{pmatrix}5\\ \:2\end{pmatrix}$
The upper element is supposed to be the 1. coordinate of the upper value from the vector. I am not quite sure what that means.
The vector is formed from the eigenvectors from the matrix $ A = \begin{pmatrix}8&-18\\ 3&-7\end{pmatrix}$
I have calculated the property that $A = PDP^{-1} \to \begin{pmatrix}8&-18\\ 3&-7\end{pmatrix} = \begin{pmatrix}3&2\\ \:\:\:\:\:\:1&1\end{pmatrix}\begin{pmatrix}2&0\\ \:\:\:\:\:\:0&-1^{\:}\end{pmatrix}\begin{pmatrix}1&-2\\ \:\:\:\:\:\:-1&3\end{pmatrix}$
And from this I have managed to get $A^n = PD^nP^{-1} \to \begin{pmatrix}3\cdot \:2^n+2&-3\cdot \:2^{1+n}-6\\ 2^n+1&-2^{n+1}-3\end{pmatrix} = \begin{pmatrix}3&2\\ 1&1\end{pmatrix}\begin{pmatrix}2^n&0\\ 0&-1^n\end{pmatrix}\begin{pmatrix}1&-2\\ -1&3\end{pmatrix}$
This is where I am stuck now. How am I supposed to find the "upper element"?
Thank you in advance!