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

We first diagonalize M.

Eigenvalues:
Solve |M - λI| = 0
==> λ^2 - 4λ = 0
==> λ = 0, 4.

Eigenvectors.
For λ = 0, we solve (A - 0I)v = 0:
[8 8|0]
[-4 -4|0], which reduces to

[1 1|0]
[0 0|0], yielding eigenvector v = (-1, 1)^T.
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For λ = 4, we solve (A - 4I)v = 0:
[4 8|0]
[-4 -8|0], which reduces to

[1 2|0]
[0 0|0], yielding eigenvector v = (-2, 1)^T.
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So, we take D =
[0 0]
[0 4], the diagonal matrix of eigenvalues,

and P =
[-1 -2]
[1 1], the matrix whose columns are the corresponding eigenvectors.

So, M = PDP^(-1)
==> M^n = P D^n P^(-1)

Computing this, M^n equals
[-1 -2][0 0][1 2]
[1 1][0 4^n][-1 -1] =

[0 -2*4^n][1 2]
[0 4^n][-1 -1] =

[2*4^n...2*4^n]
[-1*4^n....-1*4^n].
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I hope this helps!