$$A = \begin{pmatrix}2&2\\ -4&8\end{pmatrix}$$
Find formulas for the entries of $A^n$, where $n$ is a positive integer.
I think I know the method to solve this problem but I am only getting half of the entries correct in the final matrix. This is what I have done so far:
Eigenvalues are $6$ and $4$. Eigenvectors for those eigenvalues are $\displaystyle \begin{pmatrix}\frac{1}{2}\\ 1\end{pmatrix}$ and $\displaystyle \begin{pmatrix}1\\ 1\end{pmatrix}$ respectively.
$$P = \begin{pmatrix}\frac{1}{2}&1\\ \:\:1&1\end{pmatrix}$$ (I combined both eigenvectors)
$$P^{-1} = \begin{pmatrix}-2&2\\ \:\:2&-1\end{pmatrix}$$
$$B = \begin{pmatrix}6&0\\ \:0&4\end{pmatrix}$$
$$B^n = \begin{pmatrix}6^n&0\\ \:0&4^n\end{pmatrix}$$
$$A^n = P^{-1}B^{n}P$$
$$A^n = \begin{pmatrix}-2^n\cdot \:3^n+2^{2n+1}&-3^n\cdot \:2^{n+1}+2^{2n+1}\\ 2^n\cdot \:3^n-4^n&3^n\cdot \:2^{n+1}-4^n\end{pmatrix}$$
My top left and bottom right entries seem to be correct but my top right and bottom left entries still appear to be wrong. I have doubled checked and can't seem to find anything.
Would anyone happen to know why?
Thank you!