The matrix A is given by A = [[2, 2, -3], [2, 2, 3], [-3, 3, 3]] The matrix A has an eigenvector [[1], [-1], [1]]. Find the corresponding eigenvalue. The matrix A also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix P such that A = PDP⁻¹, where D is a diagonal matrix which is to be determined. The matrix B is such that B = QAQ⁻¹, where Q = [[4, 1, 5], [1, 4, 2], [1, 2, 1]] By using the expression PDP⁻¹ for A, find the set of eigenvalues and a corresponding set of eigenvectors for B.

Mark scheme

Marking points from the official mark scheme (May/June 2015). The mark or mark type is shown in brackets.

1. [M1A1] [[2,2,-3],[2,2,3],[-3,3,3]][[1],[-1],[1]] = [[-3],[-3],[-3]] ⇒ eigenvalue is – 3.
2. [M1A1] For λ = 4 : i j k / -2 2 -3 / -3 3 -1 ~ [[7],[7],[0]] (Or by equations)
3. [M1A1] For λ = 6 : i j k / -4 2 -3 / -3 3 -3 ~ [[-3],[-1],[-2]]
4. [A1] (For λ=6 eigenvector)
5. [B1] P = [[1,1,-3],[-1,4,-1],[1,2,-2]] (eigenvectors as columns)
6. [B1√] D = [[-3,0,0],[0,4,0],[0,0,6]]
7. [B1] B = QAQ⁻¹ = QPAP⁻¹Q⁻¹; = QPA(QP)⁻¹
8. [B1] QP = [[-2,15,-17],[-1,5,-7],[0,3,-3]] (CAO)
9. [B1] Eigenvalues are -3, 4 and 6 (same as those for A). (CAO)
10. [B1B1] Eigenvectors are: [[2],[1],[0]], [[15],[5],[3]], [[17],[7],[3]]