Solution
= Solution
For the slow mode choose $e_B=(1,\beta)^T$. The second row gives $c+s\beta=\lambda_B\beta$, so to the requested order
$$
\boxed{\beta=-\frac ca}.
$$
For the fast mode choose $e_A=(\alpha,1)^T$. The first row gives $(\lambda_A-q^2s)\alpha=cq^2$, hence
$$
\boxed{\alpha=\frac{cq^2}{a}}.
$$
Thus the conserved mode contains an order-one slaved nonconserved component, while the fast mode contains only an $O(q^2)$ conserved component.
Solved by gpt-5.6-sol high.