The matrix A is given by A = [[4, 1, -1], [-4, -1, 4], [0, -1, 5]].

Mark scheme

Marking points from the official mark scheme (May/June 2010). The mark or mark type is shown in brackets.

1. [1] (a) Multiply A by the given eigenvector: A[1, -2, -1]ᵀ = [[4, 1, -1], [-4, -1, 4], [0, -1, 5]] [1, -2, -1]ᵀ = [4(1)+1(-2)-1(-1), -4(1)-1(-2)+4(-1), 0(1)-1(-2)+5(-1)]ᵀ = [4-2+1, -4+2-4, 0+2-5]ᵀ = [3, -6, -3]ᵀ
2. [1] (a) Observe A x = λx => [3, -6, -3]ᵀ = 3[1, -2, -1]ᵀ. So the eigenvalue λ = 3.
3. [1] (b) Set up (A - λI)e = 0 for λ=4: (A - 4I)e = [[0, 1, -1], [-4, -5, 4], [0, -1, 1]]e = 0
4. [1] (b) Solve (A - 4I)e = 0: From R3: -e₂ + e₃ = 0 => e₃ = e₂. From R1: e₂ - e₃ = 0 (consistent). From R2: -4e₁ - 5e₂ + 4e₃ = 0 => -4e₁ - 5e₂ + 4e₂ = 0 => -4e₁ - e₂ = 0 => e₂ = -4e₁. Let e₁ = t, then e₂ = -4t, e₃ = -4t. Choose t=0 (e.g. [0, 1, 1]ᵀ or multiple). The eigenvector is [0, 1, 1]ᵀ.
5. [1] (c) Diagonal matrix D of eigenvalues: D = [[1, 0, 0], [0, 3, 0], [0, 0, 4]]. The order of eigenvalues must match the order of eigenvectors in P.
6. [1] (c) Matrix P contains eigenvectors as columns: P = [[1, 1, 0], [-4, -2, 1], [-1, -1, 1]]. (Order must match D.)
7. [1] (c) Matrix Q is the inverse of P: Q = P⁻¹
8. [2] (c) Calculate P⁻¹: For any valid method (e.g., adj(P)/det(P)). P⁻¹ = [[-2/3, -1/2, 1/3], [-1/3, 1/2, 0], [0, 0, -1/3]]. (A2 for fully correct, A1 for one error)
9. [1] (c) Conclusion about A⁵: A⁵ = PDP⁻¹.