Let $$M = \begin{bmatrix} -7 & 8 \\ -8 & -7 \end{bmatrix}.$$ Find formulas for the entries of $M^n$ where $n$ is a positive integer. (Your formulas should not contain complex numbers.) Your answer should be in the form of a matrix.
I diagonalized to the form $M = P D P^{-1}$ and $M^n = P D^n P^{-1}$ where $P$ is my matrix of eigenvectors and $D$ is my matrix of eigenvalues.
My final answer after diagonalization was $$M^n = \begin{bmatrix} > .5((-7+8i)^n+(-7-8i)^n) & (i/2)(-(-7+8i)^n+(-7-8i)^n) \\ >(.5/i)(-(-7+8i)^n+(-7-8i)^n) & .5((-7+8i)^n+(-7-8i)^n) \end{bmatrix}$$
I had a hard time finding an answer not in terms of complex numbers. Can someone show me what I'm missing?
I asked this question here a couple of days ago and I finally had time to correct it. Can someone check my solution, Every value I can think of works for it but it was an online homework assignment and my solution isn't being accepted.
After using De Moivre's formula as BaronVT suggested I came up with the solution
$$M^n = \begin{bmatrix} \sqrt{113^n}\cdot\cos\left(n\cdot\left(\tan^{-1}\left(\frac{8}{-7}\right)+\pi\right)\right) & \sqrt{113^n}\cdot\sin\left(n\cdot\left(\tan^{-1}\left(\frac{8}{-7}\right)+\pi\right)\right) \\ -\sqrt{113^n}\cdot\sin\left(n\cdot\left(\tan^{-1}\left(\frac{8}{-7}\right)+\pi\right)\right) & \sqrt{113^n}\cdot\cos\left(n\cdot\left(\tan^{-1}\left(\frac{8}{-7}\right)+\pi\right)\right) \end{bmatrix}$$