Find formulas for the entries of M^n

[icefall5]

Homework Statement

Let $M = \begin {bmatrix} 8 & -1 \\ 2 & 11 \\ \end{bmatrix}$

Find formulas for the entries of $M^n$, where n is a positive integer.

Homework Equations

N/A

The Attempt at a Solution

I honestly have no clue where to start. We recently covered diagonalization, but I can't see how this relates.

 

[clamtrox]

Let's suppose D is the transform that diagonalizes M, and denote $D M D^{-1} = \tilde{M}$. Then
$$M^n = M M M ... M = D^{-1} D M D^{-1} D M D^{-1} D M ... M D^{-1} D = D^{-1} \tilde{M}^n D.$$

 

[HallsofIvy]

Let's suppose D is the transform that diagonalizes M, and denote $D M D^{-1} = \tilde{M}$. Then
$$M^n = M M M ... M = D^{-1} D M D^{-1} D M D^{-1} D M ... M D^{-1} D = D^{-1} \tilde{M}^n D.$$

Also, The diagonal entries in $\tilde{M}$ are the eigenvalues of the matrix M and the columns of D are the corresponding eigenvectors of M.