Find the eigenvalues and corresponding eigenvectors of the matrix \( A = \begin{pmatrix} 4 & -1 & 1 \\ -1 & 0 & -3 \\ 1 & -3 & 0 \end{pmatrix} \).

Mark scheme

Marking points from the official mark scheme (May/June 2011). The mark or mark type is shown in brackets.

1. [M1A1] Finds characteristic equation and solves. Det(A – λ I) = 0 ⇒ λ³ - 4λ² - 11λ + 30 = 0 ⇒ λ = -3, 2, 5 (Any one)
2. [A1] (All three)
3. [M1A1] Uses vector product (or equations) to find corresponding eigenvectors. \( \lambda = -3 \quad \mathbf{e}_1 = \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} \)
4. [A1] \( \lambda = 2 \quad \mathbf{e}_2 = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix} \)
5. [A1] \( \lambda = 5 \quad \mathbf{e}_3 = \begin{pmatrix} 1 \\ -1 \\ -1 \end{pmatrix} \)
6. [B1\( \checkmark \)] Forms P from eigenvectors and D with fifth powers of eigenvalues on leading diagonal.
7. [M1] Note: Columns of P and D can be permuted, but must match. \( \mathbf{P} = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 1 & -1 \\ 1 & -1 & -1 \end{pmatrix} \)
8. [A1\( \checkmark \)] \( \mathbf{D} = \begin{pmatrix} (-3)^5 & 0 & 0 \\ 0 & 2^5 & 0 \\ 0 & 0 & 5^5 \end{pmatrix} = \begin{pmatrix} -243 & 0 & 0 \\ 0 & 32 & 0 \\ 0 & 0 & 3125 \end{pmatrix} \)