Edexcel A-Level Mathematics A2 Fp3 6 8 Reduction of Symmetric Matrices Questions

Practise pairing eigenvalues with normalised eigenvectors, constructing orthogonal P and diagonal D, and verifying D = PᵀMP from local FP3 evidence.

Exam points

Place eigenvalues in D in the same order as the normalised eigenvector columns of P.
Use orthogonal eigenvectors to construct P and verify the form D=P^T M P.

[Maximum number: 4]

\mathbf{M}=\left(\begin{array}{rrr} 0 & -1 & 3 \\ -1 & 4 & -1 \\ 3 & -1 & 0 \end{array}\right)

Given that $\left(\begin{array}{r}1 \\ -2 \\ 1\end{array}\right)$ is an eigenvector of M

Hence, given that $\begin{pmatrix}1\\1\\1\end{pmatrix}$ is also an eigenvector of $\mathbf{M}$

determine a diagonal matrix D and an orthogonal matrix P such that $\mathbf{D}=\mathbf{P}^{\mathrm{T}} \mathbf{M P}$

\lambda=2

\mathbf D=\begin{smallmatrix}-3&0&0\\0&2&0\\0&0&5\end{smallmatrix}

\begin{smallmatrix}1\\-2\\1\end{smallmatrix}\to\begin{smallmatrix}\frac{\sqrt6}{6}\\-\frac{\sqrt6}{3}\\\frac{\sqrt6}{6}\end{smallmatrix},\quad \begin{smallmatrix}1\\0\\-1\end{smallmatrix}\to\begin{smallmatrix}\frac{\sqrt2}{2}\\0\\-\frac{\sqrt2}{2}\end{smallmatrix},\quad \begin{smallmatrix}1\\1\\1\end{smallmatrix}\to\begin{smallmatrix}\frac{\sqrt3}{3}\\\frac{\sqrt3}{3}\\\frac{\sqrt3}{3}\end{smallmatrix}

\mathbf P=\begin{smallmatrix}\frac{\sqrt2}{2}&\frac{\sqrt3}{3}&\frac{\sqrt6}{6}\\0&\frac{\sqrt3}{3}&-\frac{\sqrt6}{3}\\-\frac{\sqrt2}{2}&\frac{\sqrt3}{3}&\frac{\sqrt6}{6}\end{smallmatrix}

B1: correct remaining eigenvalue 2.\nB1ft: diagonal matrix consistent with eigenvalues.\nM1: normalises at least one eigenvector.\nA1: fully correct consistent $\mathbf D$ and $\mathbf P$ .

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Edexcel A-Level Mathematics A2 Fp3 6 8 Reduction of Symmetric Matrices Questions question 1