# Diagonalizing and expressing as A^k

## Homework Statement

Let $$A=\left(\begin{array}{cc}11 & 6\\-12 & -6\end{array}\right)$$ and let k be a positive integer. By diagonalizing A, express Ak as an explicit 2x2 matrix in terms of k. Your answer should take the form $$A=\left(\begin{array}{cc}f_1 (k) & f_2 (k)\\f_3 (k) & f_4 (k)\end{array}\right)$$

## Homework Equations

P-1AP=D and Ak=PDkP-1.

## The Attempt at a Solution

I don't have a problem diagonalizing, I have a problem with expressing it in terms of functions (at least I think). I diagonalized normally and got $$P=\left(\begin{array}{cc}-3 & -2\\4 & 3\end{array}\right)$$ and $$D=\left(\begin{array}{cc}3 & 0\\0 & 2\end{array}\right)$$ After using the second formula above, I get $$A^k=\left(\begin{array}{cc}9*3^k-8*2^k & 6*3^k-6*2^k\\-12*3^k+12*2^k & -8*3^k+9*2^k\end{array}\right)$$ I am just not sure if this is "good enough" or even right? Is there some way I can combine this? Just want to double check my answer basically

micromass
Staff Emeritus
Homework Helper
This looks good!!

Thank you