I have to find a formula given $a_n$ using the following information: $M^k =PD^k P^{-1}$
I have been given the following information: $a_1= -1$, $a_2= 1$
The matrix $M$ is $ \begin{bmatrix} 0 & 1 \\-20 & 9 \\ \end{bmatrix} $.
Furthermore, the exponent $k$ such that $\begin{bmatrix} a_n \\a_{n+1} \\ \end{bmatrix}$ $=$ $M^k$ $\begin{bmatrix} a_1 \\a_2 \\ \end{bmatrix}$
Here is where I begin my work:
When I diagonalize the matrix, $D= \begin{bmatrix} 5 & 0 \\0 & 4 \\ \end{bmatrix}$ , $P=\begin{bmatrix} 1 & 1 \\5 & 4 \\ \end{bmatrix}$ and $P^{-1} =\begin{bmatrix} -4 & 1 \\5 & -1 \\ \end{bmatrix} $
Since $M^k =PD^k P^{-1}$ and
$\begin{bmatrix} a_n \\a_{n+1} \\ \end{bmatrix}$ $=$ $M^k$ $\begin{bmatrix} a_1 \\a_2 \\ \end{bmatrix}$, where $k = n-1$.
Here's where I start my work. I wrote the following:
$\begin{bmatrix} 1 & 1 \\ 5& 4 \\ \end{bmatrix}$$\begin{bmatrix} 5^k & 0^k \\ 0^k & 4^k \\ \end{bmatrix}$$\begin{bmatrix} -4 & 1 \\ 5 & -1 \\ \end{bmatrix}$ $\begin{bmatrix} -1 \\ 1 \\ \end{bmatrix}$
which solves to:
$\begin{bmatrix} a_n \\a_{n+1} \\ \end{bmatrix}$= $\begin{bmatrix} 5^{k+1} & -6*4^k \\ 5^{k+2}& -6*4^{k+1} \\ \end{bmatrix}$
Since $k = n-1$. I concluded that the $a_n$ formula to be:
$5^n - 6*4 ^{n-1}$
However it shows that I got this formula wrong. Can somebody lend a hand?