= Solution
Try a <particular integral> along each forcing <eigenvector>. Since $A\mathbf a_j=\lambda_j\mathbf a_j$,
$$
(\partial_t-A)\left(\frac{\mathbf a_j e^{i\omega_jt}}{i\omega_j-\lambda_j}\right)=\mathbf a_j e^{i\omega_jt}.
$$
With both denominators nonzero, add the two responses:
$$
\boxed{\mathbf x_p=\frac{\mathbf a_1e^{i\omega_1t}}{i\omega_1-\lambda_1}
+\frac{\mathbf a_2e^{i\omega_2t}}{i\omega_2-\lambda_2}.}
$$
Only the forcing component's own <eigenvalue> enters its denominator; a frequency matching a different eigenmode does not create resonance in this already diagonal forcing component.
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