CAIE A-Level Further Math A2 2.2.5 Eigenvalues and Eigenvectors Questions

$\lambda=-6, \quad \lambda=-2, \quad \lambda=8$$\lambda=-6:\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} 0 4 0 0 2\end{array}\right|=\left(\begin{array}{l}8 0 0\end{array}\right) \sim\left(\begin{array}{l}1 0 0\end{array}\right)$
Uses vector product (or equations) to find corresponding
eigenvectors.

$\lambda=-2:\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} -4 2 0 0 5\end{array}\right|=\left(\begin{array}{c}10 20 0\end{array}\right) \sim\left(\begin{array}{c}1 2 0\end{array}\right)$$\lambda=8:\left|\begin{array}{ccc}\mathbf{i} \mathbf{j} \mathbf{k} -14 2 0 -10 5\end{array}\right|=\left(\begin{array}{c}140 70 140\end{array}\right) \sim\left(\begin{array}{l}2 1 2\end{array}\right)$
Thus $\mathbf{P}=\left(\begin{array}{lll}1 1 0 2 0 0 2\end{array}\right)$ and $\mathbf{D}=\left(\begin{array}{ccc}-\frac{1}{6} 0 0 -\frac{1}{2} 0 0 \frac{1}{8}\end{array}\right)$
Or correctly matched permutations of columns.
M1 for their (non-zero) eigenvectors matched to their
eigenvalues.

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