= Solution
Let $\kappa=T/(2ma)$ and seek a <normal mode> $\mathbf y=\mathbf a e^{i\omega t}$. Its <eigenvalue equation> is
$$
\det\begin{pmatrix}3-\omega^2/\kappa&-2\\-1&2-\omega^2/\kappa\end{pmatrix}=0,
$$
or $\lambda^2-5\lambda+4=(\lambda-1)(\lambda-4)=0$, where $\lambda=\omega^2/\kappa$. Thus
$$
\boxed{\omega_-=\sqrt{\frac{T}{2ma}},\qquad
\omega_+=2\sqrt{\frac{T}{2ma}}.}
$$
These are <angular frequencies>; the corresponding ordinary <frequencies> are $f_\pm=\omega_\pm/(2\pi)$.
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