= Solution
A <normal mode> $x\propto e^{-i\omega t}$ has eigenvalue $\lambda=-i\omega$ of $A$. Setting $d_\omega=\omega-\omega_1$, the characteristic equation is
$$
(-id_\omega)(-i(d_\omega+b))-ac=0,
\qquad d_\omega^2+bd_\omega+ac=0.
$$
Thus
$$
\omega=\omega_1-\frac b2\pm\sqrt{\frac{b^2}{4}-ac},\qquad
\boxed{\text{exponential modal instability}\ \Longleftrightarrow\ 4ac>b^2.}
$$
One branch then has positive imaginary frequency and hence positive exponential <growth rate>. If $4ac<b^2$, both frequencies are real and the distinct-eigenvalue system has bounded oscillatory motion. At equality there is no positive exponential growth rate, but a nontrivial <Jordan block> can cause algebraic growth, so exponential neutrality must not be confused with <Lyapunov stability>. The parameter $d$ listed in the PDF does not appear in the printed matrix and has no effect on this calculation.
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