CAIE A-Level Further Mathematics 2.2.5 Eigenvalues and Eigenvectors

Practise finding matched eigenvectors, arranging them as columns of a modal matrix and transforming the eigenvalues correctly for powers, inverses or shifted matrices.

Exam points

solve for a non-zero eigenvector for each eigenvalue and place these vectors as columns of P
match every column of P to the corresponding diagonal entry before transforming that eigenvalue
use the requested power, inverse or shift on the diagonal entries while retaining the same eigenvectors

[Maximum number: 5]

It is given that a is a positive constant.

Find a matrix P such that

\mathbf{A}=\mathbf{P}\left(\begin{array}{rrr} a & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -4 \end{array}\right) \mathbf{P}^{-1} .

$\lambda=a:\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} 0 -4-a 0 3 -1-a\end{array}\right|=\left(\begin{array}{c}(-4-a)(-1-a) 0 0\end{array}\right) \sim\left(\begin{array}{l}1 0 0\end{array}\right)$

M1 A1

Uses vector product (or equations) to find
corresponding eigenvectors.

$\lambda=-1: \quad\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} a+1 2 a+5 a+1 0 -3 0\end{array}\right|=\left(\begin{array}{c}3(a+1) 0 -3(a+1)\end{array}\right) \sim\left(\begin{array}{c}1 0 -1\end{array}\right)$

A1

$\lambda=-4:\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} a+4 2 a+5 a+1 0 3 3\end{array}\right|=\left(\begin{array}{c}3 a+12 -(3 a+12) 3 a+12\end{array}\right) \sim\left(\begin{array}{c}1 -1 1\end{array}\right)$

A1

$\mathbf{P}=\left(\begin{array}{ccc}1 1 0 0 -1 0 -1 1\end{array}\right)$

A1

OE

5

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2.2.5—Eigenvalues and eigenvectors question 1