The matrix A, where A = (1 0 0 10 -7 10 7 -5 8) has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that (0 2 1)ᵀ is an eigenvector of A. Find the corresponding eigenvalue. Find a diagonal matrix D and matrices P and P⁻¹ such that P⁻¹AP = D.

Mark scheme

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

1. [M1] (a) Attempt to find eigenvector for λ=1 (solve (A-I)v=0)
2. [A1] (a) Correct eigenvector for λ=1 (e.g., (1 1 0)ᵀ)
3. [A1] (a) Correct eigenvector for λ=3 (e.g., (0 1 1)ᵀ)
4. [M1] (b) Multiply A by the given eigenvector (0 2 1)ᵀ
5. [A1] (b) Identify the scalar multiple to find the eigenvalue: -2
6. [B1] (c) Correct diagonal matrix D (eigenvalues on diagonal in any order)
7. [B1] (c) Correct matrix P (corresponding eigenvectors in same order as D)
8. [B1] (c) Correct determinant of P
9. [M1] (c) Attempt to find inverse of P (e.g., adj(P)/det(P) or row operations)
10. [A1] (c) Correct P⁻¹