Pearson Edexcel IAL Mathematics FP3.6.7 Eigenvalues & eigenvectors

Practise finding eigenvalues and eigenvectors of 2 × 2 and 3 × 3 matrices, including normalised vectors where required.

Exam points

use Mv = λv or det(M - λI) to determine missing eigenvalues or parameters
solve (M - λI)x = 0 to find a corresponding eigenvector for a given eigenvalue
normalise an eigenvector and state it in vector form when the question requires it

[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

(a)

determine its corresponding eigenvalue.

[ 2 ]

\mathbf M\begin{smallmatrix}1\\-2\\1\end{smallmatrix}=\begin{smallmatrix}5\\-10\\5\end{smallmatrix}=5\begin{smallmatrix}1\\-2\\1\end{smallmatrix}

\lambda=5

M1: correct method leading to a value for $\lambda$ .\nA1: correct value.

(b)

determine a corresponding eigenvector.

[ 2 ]

(\mathbf M+3\mathbf I)\begin{smallmatrix}x\\y\\z\end{smallmatrix}=\begin{smallmatrix}0\\0\\0\end{smallmatrix}

\begin{smallmatrix}3&-1&3\\-1&7&-1\\3&-1&3\end{smallmatrix}\begin{smallmatrix}x\\y\\z\end{smallmatrix}=\begin{smallmatrix}0\\0\\0\end{smallmatrix}

A corresponding eigenvector is

k\begin{smallmatrix}1\\0\\-1\end{smallmatrix}

where $k\ne0$ .\nM1: uses $\mathbf M\mathbf x=-3\mathbf x$ or $(\mathbf M+3\mathbf I)\mathbf x=0$ to obtain a non-zero vector.\nA1: $k(1,0,-1)^T$ .

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