Write down the eigenvalues of the matrix A, where A = [[-2, 1, -1], [0, -1, 2], [0, 0, 1]] and find corresponding eigenvectors. Find a matrix P and a diagonal matrix D such that P⁻¹AP = D, and hence find the matrix Aⁿ, where n is a positive integer.

Mark scheme

Marking points from the official mark scheme (May/June 2016). The mark or mark type is shown in brackets.

1. [B1] Eigenvalues are:-2, -1, 1
2. [M1A1A1] Eigenvectors are; (1,0,0), (1,1,0), (0, -1, 1) (or equivalent)
3. [B1B1] P = [[1,1,0],[0,1,1],[0,0,1]] , D = [[-2,0,0],[0,-1,0],[0,0,1]] (or equivalent in correct order)
4. [M1A1] P⁻¹ = [[1,-1,1],[0,1,-1],[0,0,1]]
5. [M1] (P⁻¹AP)ⁿ = P⁻¹AⁿP = Dⁿ
6. [A1] Dⁿ = [[(-2)ⁿ,0,0],[0,(-1)ⁿ,0],[0,0,1]]
7. [M1] Aⁿ = P Dⁿ P⁻¹
8. [A1] Aⁿ = [[(-2)ⁿ, -(-2)ⁿ+(-1)ⁿ, -(-2)ⁿ+(-1)ⁿ+1],[0, (-1)ⁿ, (-1)ⁿ+1],[0,0,1]]