Question 1170702: Let M=[{0,-2},{4,6}]. Find formulas for the entries of Mn, where n is a positive integer.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's find the formula for the entries of $M^n$, where $M = \begin{bmatrix} 0 & -2 \\ 4 & 6 \end{bmatrix}$.
**1. Find the Eigenvalues and Eigenvectors of M**
* **Characteristic Equation:**
det(M - λI) = 0
det( $\begin{bmatrix} -\lambda & -2 \\ 4 & 6-\lambda \end{bmatrix}$ ) = 0
(-λ)(6 - λ) - (-2)(4) = 0
-6λ + λ² + 8 = 0
λ² - 6λ + 8 = 0
(λ - 4)(λ - 2) = 0
* **Eigenvalues:**
λ₁ = 4 and λ₂ = 2
* **Eigenvectors:**
* For λ₁ = 4:
(M - 4I)v₁ = 0
$\begin{bmatrix} -4 & -2 \\ 4 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$
-4x - 2y = 0
-2x - y = 0
y = -2x
Let x = 1, then y = -2.
v₁ = $\begin{bmatrix} 1 \\ -2 \end{bmatrix}$
* For λ₂ = 2:
(M - 2I)v₂ = 0
$\begin{bmatrix} -2 & -2 \\ 4 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$
-2x - 2y = 0
x + y = 0
y = -x
Let x = 1, then y = -1.
v₂ = $\begin{bmatrix} 1 \\ -1 \end{bmatrix}$
**2. Form the Matrix P and Diagonal Matrix D**
* P = $\begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix}$ (matrix of eigenvectors)
* D = $\begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix}$ (diagonal matrix of eigenvalues)
**3. Find P⁻¹**
* det(P) = (1)(-1) - (1)(-2) = -1 + 2 = 1
* P⁻¹ = (1/det(P)) $\begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$ = $\begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$
**4. Use the Formula Mⁿ = PDⁿP⁻¹**
* Dⁿ = $\begin{bmatrix} 4ⁿ & 0 \\ 0 & 2ⁿ \end{bmatrix}$
* Mⁿ = $\begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix} \begin{bmatrix} 4ⁿ & 0 \\ 0 & 2ⁿ \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$
* Mⁿ = $\begin{bmatrix} 4ⁿ & 2ⁿ \\ -2(4ⁿ) & -2ⁿ \end{bmatrix} \begin{bmatrix} -1 & -1 \\ 2 & 1 \end{bmatrix}$
* Mⁿ = $\begin{bmatrix} -4ⁿ + 2(2ⁿ) & -4ⁿ + 2ⁿ \\ 2(4ⁿ) - 2(2ⁿ) & 2(4ⁿ) - 2ⁿ \end{bmatrix}$
* Mⁿ = $\begin{bmatrix} -4ⁿ + 2^(n+1) & -4ⁿ + 2ⁿ \\ 2(4ⁿ) - 2^(n+1) & 2(4ⁿ) - 2ⁿ \end{bmatrix}$
**Formulas for the Entries:**
* Mⁿ₁₁ = -4ⁿ + 2^(n+1)
* Mⁿ₁₂ = -4ⁿ + 2ⁿ
* Mⁿ₂₁ = 2(4ⁿ) - 2^(n+1)
* Mⁿ₂₂ = 2(4ⁿ) - 2ⁿ
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