Write down the eigenvalues of the matrix A, where A = [[1, 4, -16], [0, 2, 3], [0, 0, 3]]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2012). The mark or mark type is shown in brackets.

1. [B1] States eigenvalues: Eigenvalues are 1, 2, 3.
2. [M1A1] Finds eigenvectors for λ=1: (A-I)e₁=0 ⇒ [[0, 4, -16], [0, 1, 3], [0, 0, 2]] e₁ = 0 ⇒ e₁ = (28, -2, 0)ᵀ
3. [A1] Finds eigenvectors for λ=2: (A-2I)e₂=0 ⇒ [[-1, 4, -16], [0, 0, 3], [0, 0, 1]] e₂ = 0 ⇒ e₂ = (4, 1, 0)ᵀ
4. [A1] Finds eigenvectors for λ=3: (A-3I)e₃=0 ⇒ [[-2, 4, -16], [0, -1, 3], [0, 0, 0]] e₃ = 0 ⇒ e₃ = (-2, 3, 1)ᵀ
5. [B1] States P: P = [[28, 4, -2], [-2, 1, 3], [0, 0, 1]]
6. [B1] States D: D = [[1, 0, 0], [0, 2, 0], [0, 0, 3]]
7. [M1A1] Finds inverse of P: Det P = 1. Adj P = P⁻¹ = [[1, -4, 14], [0, 1, -3], [0, 0, 1]]
8. [M1A1] Finds Aⁿ: Aⁿ = PDP⁻¹ = P Dⁿ P⁻¹ = [[28, 4, -2], [-2, 1, 3], [0, 0, 1]] [[1ⁿ, 0, 0], [0, 2ⁿ, 0], [0, 0, 3ⁿ]] [[1, -4, 14], [0, 1, -3], [0, 0, 1]]
9. [A1] Aⁿ result: [[1, -4+4*2ⁿ, 14-12*2ⁿ-2*3ⁿ], [0, 2ⁿ, -3*2ⁿ+3ⁿ], [0, 0, 3ⁿ]]
10. [B1] States required limit: lim (as n→∞) (3⁻ⁿAⁿ) = [[0, 0, 0], [0, 0, 0], [0, 0, 1]]