Linear algebra help

Question

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

Answer 1

Mⁿ = PDⁿP^-1

Where Dⁿ = [e₁ⁿ 0]

[0 e₂ⁿ]

The eigenvalues of M are 8 and 3

Dⁿ=[8ⁿ 0]

[0 3ⁿ]

P = [1 0.5]

[1 1]

Answer 2

Diagonalize M::

The eigenvalues of M are 3 and 8.

An eigenvector corresponding to the eigenvalue 3 is <1,2> and an eigenvector corresponding to the eigenvalue 8 is <1,1>.

Let P be the 2x2 matrix with first column the eigenvector for 3 and the second column the eigenvector for 8.

Then, P^-1MP is the 2x2 diagonal matrix with the eigenvaliues 3 and 8 on the main diagonal (in that order).

Let A = P^-1MP. Then, Aⁿ = P^-1MⁿP. So, Mⁿ = PAⁿP^-1. Since A is a 2x2 diagonal matrix with the entries 3 and 8 on the main diagonal, Aⁿ will also be a 2x2 diagonal matrix with 3ⁿ and 8ⁿ as the main diagonal entries (in that order).

Do the multiplication to get formulas for the entries of Mⁿ.