Simplify the Matrix [[12,108],[4,-12]]^n

Problem

[[12,108],[4,−12]]n

Solution

1. Identify the matrix A and factor out the greatest common divisor of the entries to simplify calculations.

A=[[12,108],[4,−12]]=4*[[3,27],[1,−3]]

2. Calculate the eigenvalues of the inner matrix B=[[3,27],[1,−3]] by solving the characteristic equation det(B−λ*I)=0

det(3−λ)=(3−λ)*(−3−λ)−27=0

λ2−9−27=0

λ2=36

λ=±6

3. Find the eigenvectors for (λ_1)=6 and (λ_2)=−6
   For (λ_1)=6

[[−3,27],[1,−9]]*[[x],[y]]=[[0],[0]]⇒x=9*y⇒(v_1)=[[9],[1]]

For (λ_2)=−6

[[9,27],[1,3]]*[[x],[y]]=[[0],[0]]⇒x=−3*y⇒(v_2)=[[−3],[1]]

4. Diagonalize the matrix B using B=P*D*P(−1) where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues.

P=[[9,−3],[1,1]]

D=[[6,0],[0,−6]]

P(−1)=1/12*[[1,3],[−1,9]]

5. Apply the power rule for matrices An=(4*B)n=4*Bn=4*P*Dn*P(−1)

Dn=[[6,0],[0,(−6)n]]=6*[[1,0],[0,(−1)n]]

An=4⋅6*[[9,−3],[1,1]]*[[1,0],[0,(−1)n]]1/12*[[1,3],[−1,9]]

An=24/12*[[9,−3*(−1)n],[1,(−1)n]]*[[1,3],[−1,9]]

6. Multiply the remaining matrices to find the general form.

An=24/12*[[9+3*(−1)n,27−27*(−1)n],[1−(−1)n,3+9*(−1)n]]

Final Answer

[[12,108],[4,−12]]n=24/12*[[9+3*(−1)n,27−27*(−1)n],[1−(−1)n,3+9*(−1)n]]

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