= Solution
For the modal <matrix> $M(s)$, $s\ge0$, its <trace> is always negative:
$$
\operatorname{tr}M(s)=-(1+\kappa)-(\alpha+1)s<0.
$$
Its <determinant> is
$$
D(s)=(1+\alpha s)(\kappa+s)-\beta\kappa\eta s
=\alpha s^2+(1+\alpha\kappa-\beta\kappa\eta)s+\kappa.
$$
The proposed inequality is exactly $\beta\kappa\eta<\alpha\kappa+1$. It makes the coefficient of $s$ positive, while the leading and constant coefficients are already positive. Therefore $D(s)>0$ for every $s\ge0$, independent of which discrete wave numbers the domain allows.
A real two-by-two <matrix> with negative <trace> and positive <determinant> has both <eigenvalues> with negative <real part>: if they are real they have positive product and negative sum, and if <nonreal> their common <real part> is half the <trace>. Every mode therefore decays, proving \b[<linear stability> for all domain sizes] under the stated sufficient condition.
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