FP3 June 2014 (R) Q6
6. The symmetric matrix \(\mathbf{M}\) has eigenvectors \(\begin{pmatrix}2 \\ 2 \\ 1\end{pmatrix},\ \begin{pmatrix}-2 \\ 1 \\ 2\end{pmatrix}\) and \(\begin{pmatrix}1 \\ -2 \\ 2\end{pmatrix}\) with eigenvalues 5, 2 and \(-1\) respectively.
(a) Find an orthogonal matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \[\mathbf{P}^{\mathrm{T}}\mathbf{M}\mathbf{P} = \mathbf{D}\] (4)
Given that \(\mathbf{P}^{-1} = \mathbf{P}^{\mathrm{T}}\)
(b) show that \[\mathbf{M} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\] (2)
(c) Hence find the matrix \(\mathbf{M}\). (5)
| Scheme | Marks |
|---|---|
| \(\mathbf{P} = \begin{pmatrix}\dfrac{2}{3} & -\dfrac{2}{3} & \dfrac{1}{3} \\ \dfrac{2}{3} & \dfrac{1}{3} & -\dfrac{2}{3} \\ \dfrac{1}{3} & \dfrac{2}{3} & \dfrac{2}{3}\end{pmatrix} = \left(\dfrac{1}{3}\begin{pmatrix}2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2\end{pmatrix}\right)\) M1: Attempt unit eigenvectors A1: Correct matrix | M1A1 |
| \(\mathbf{D} = \begin{pmatrix}5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{pmatrix}\) M1: Correct form for \(\mathbf{D}\) with eigenvalues in the diagonal A1: Consistent with \(\mathbf{P}\) | M1A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\mathbf{M}\mathbf{P} = \mathbf{P}\mathbf{D}\) or \(\mathbf{P}^{-1}\mathbf{M} = \mathbf{D}\mathbf{P}^{-1}\) | M1 |
| \(\mathbf{M} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\) | A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{P}^{-1} = \mathbf{P}^{\mathrm{T}} = \dfrac{1}{3}\begin{pmatrix}2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2\end{pmatrix}\) Correct matrix. Allow the transpose of their \(\mathbf{P}\). | B1ft |
| \(\mathbf{PD} = \dfrac{1}{3}\begin{pmatrix}2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2\end{pmatrix}\begin{pmatrix}5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{pmatrix} = \dfrac{1}{3}\begin{pmatrix}10 & -4 & -1 \\ 10 & 2 & 2 \\ 5 & 4 & -2\end{pmatrix}\) or \(\mathbf{DP}^{-1} = \begin{pmatrix}5 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -1\end{pmatrix}\dfrac{1}{3}\begin{pmatrix}2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2\end{pmatrix} = \dfrac{1}{3}\begin{pmatrix}10 & 10 & 5 \\ -4 & 2 & 4 \\ -1 & 2 & -2\end{pmatrix}\) M1: Attempt \(\mathbf{PD}\) or \(\mathbf{DP}^{-1}\) where \(\mathbf{D} \neq k\mathbf{I}\) A1: Correct matrix | M1A1 |
| \(\mathbf{M} = \dfrac{1}{9}\begin{pmatrix}10 & -4 & -1 \\ 10 & 2 & 2 \\ 5 & 4 & -2\end{pmatrix}\begin{pmatrix}2 & 2 & 1 \\ -2 & 1 & 2 \\ 1 & -2 & 2\end{pmatrix} = \begin{pmatrix}3 & 2 & 0 \\ 2 & 2 & 2 \\ 0 & 2 & 1\end{pmatrix}\) Or \(\mathbf{M} = \dfrac{1}{9}\begin{pmatrix}2 & -2 & 1 \\ 2 & 1 & -2 \\ 1 & 2 & 2\end{pmatrix}\begin{pmatrix}10 & 10 & 5 \\ -4 & 2 & 4 \\ -1 & 2 & -2\end{pmatrix} = \begin{pmatrix}3 & 2 & 0 \\ 2 & 2 & 2 \\ 0 & 2 & 1\end{pmatrix}\) M1: Completes correctly to find \(\mathbf{M}\) A1: Correct \(\mathbf{M}\) | M1A1 |
| Failure to use unit eigenvectors in (a) could score M0A0M1A1 in (a) and B1ftM1A0M1A0 in (c) | |
| (5) | |
| (11 marks) |