Find a formula in terms of k for the entries of A^k, where A is the diagonalizable matrix below.
A= 5 -16
2 -7
5.....-16
2.....-7
4-2(-3)^k.....-8+8(-3)^k
1-(-3)^k......-2+4(-3)^k
1/2 of all that
To find a formula for the entries of A^k, we need to diagonalize matrix A.
Let's start by finding the eigenvalues λ of matrix A:
Det(A - λI) = 0, where I is the identity matrix.
The determinant is: (5-λ)(-7-λ) - (2)(-16) = λ^2 + 2λ - 3 = (λ+3)(λ-1) = 0
So the eigenvalues are λ = -3 and λ = 1.
Next, we need to find the eigenvectors corresponding to each eigenvalue.
For λ = -3:
(A - (-3)I)x = 0
⇒ 8x₁ - 16x₂ = 0
⇒ 4x₁ - 8x₂ = 0
⇒ x₁ = 2x₂
Let x₂ = t (a free variable), then x₁ = 2t.
So, the eigenvector corresponding to λ = -3 is x = [2t, t], t ≠ 0.
For λ = 1:
(A - I)x = 0
⇒ 4x₁ - 16x₂ = 0
⇒ x₁ = 4x₂
Let x₂ = s (a free variable), then x₁ = 4s.
So, the eigenvector corresponding to λ = 1 is x = [4s, s], s ≠ 0.
Now, let's find the inverse of matrix P, which is constructed by placing the eigenvectors as column vectors:
P = [2t, 4s; t, s]
To get the Diagonal matrix D, we need to find the product of P and A. Let's denote P^-1 as Q (inverse of P):
Q = P^(-1) = (1/8)[s, -4s; -t, 2t]
Now, we can calculate D = QAP:
D = QAP = (1/8)[s, -4s; -t, 2t][5, -16; 2, -7][2t, 4s; t, s]
Multiplying the matrices, we get:
D = (1/8)[s, -4s; -t, 2t][10t+8s, 20s-16t; 5t+2s, 10s-t]
= (1/8)[(10ts - 20st) + (-16ts + 40s²), (20s - 16t)² + (-4s + 10t)(20s - 16t); (-5ts + 10t²) + (2ts- 4s²), (4s - 10t)(2ts - 4s²) + (-2t + 5s)(4s - 10t)]
= (1/8)[-6ts + 40s², 144t² - 144st + 36s²; -3ts + 10t², 12t³ - 83st² + 100s³]
Therefore, the formula for the entries of A^k is:
A^k = PD^kP^-1
= P(1/8)[-6ts + 40s², 144t² - 144st + 36s²; -3ts + 10t², 12t³ - 83st² + 100s³](1/8)[s, -4s; -t, 2t]
= (1/64)[8kts - 80ks², -96k²ts + 288k²st - 72k²s² + 96k²ts - 288k²st + 72k²s² - 16k³ts + 40k³t²; 4kts - 40kt², -48k²ts + 144k²st - 36k²s² + 16k³ts - 40k³t² + 10k³t² - 20k³t³]
Simplifying further, we can write the formula for entries of A^k as:
(A^k)₁₁ = (1/64)[8kts - 80ks² - 96k²s² + 16k³ts + 40k³t²]
(A^k)₁₂ = (1/64)[-96k²ts + 288k²st - 72k²s² - 16k³ts + 40k³t²]
(A^k)₂₁ = (1/64)[4kts - 40kt² - 48k²s² + 16k³ts - 40k³t² + 10k³t²]
(A^k)₂₂ = (1/64)[-48k²ts + 144k²st - 36k²s² + 16k³ts - 40k³t² + 10k³t² - 20k³t³]
Therefore, the formula for the entries of A^k in terms of k is:
(A^k)₁₁ = (1/64)(8kts - 80ks² - 96k²s² + 16k³ts + 40k³t²)
(A^k)₁₂ = (1/64)(-96k²ts + 288k²st - 72k²s² - 16k³ts + 40k³t²)
(A^k)₂₁ = (1/64)(4kts - 40kt² - 48k²s² + 16k³ts - 40k³t² + 10k³t²)
(A^k)₂₂ = (1/64)(-48k²ts + 144k²st - 36k²s² + 16k³ts - 40k³t² + 10k³t² - 20k³t³)
So, these are the formulas for the entries of A^k in terms of k using the diagonalizable matrix A.