Dear Lingling,
Your understanding of how the natural frequencies are derived from the state matrix is correct. For further information on the mathematical background, I recommend reviewing a textbook that covers linear state-space-based models and their solution via the matrix exponential (e.g., a modern controls textbook).
The eignevalues of the state matrix contain information on the natural frequencies and damping ratios of the system.
As described in this forum topic: Learizing Baseline 5MW Wind Turbine with FAST, rigid-body modes show up in MBC3 as a pair of zero-valued (or near-zero-valued) frequencies with +/- inf damping (i.e., eigenvalues with real values only). You should confirm with MSC, but I suspect the ADAMS linearization produces something similar.
Best regards,