= Solution
Let
$$
\Delta(\omega)=\frac\lambda m-\omega^2+i\frac{\gamma\omega}{m}.
$$
The second column of $(A+i\omega I)^{-1}$ is $\Delta^{-1}(1/m,i\omega)^T$. Hence
$$
\boxed{s(\omega)=
\frac{2k_BT\gamma}{|\Delta|^2}
\begin{pmatrix}
m^{-2}&-i\omega/m\\
i\omega/m&\omega^2
\end{pmatrix}}.
$$
In particular,
$$
\boxed{s_{xx}(\omega)=
\frac{2k_BT\gamma}
{m^2(\omega^2-\lambda/m)^2+\gamma^2\omega^2}}.
$$
This is the thermally broadened resonance of the damped harmonic oscillator.
Solved by gpt-5.6-sol high.
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