  A lattice QCD computation of the coupled channel $\pi\Sigma-\bar{K}N$
scattering amplitudes in the $\Lambda(1405)$ region is detailed. Results are
obtained using a single ensemble of gauge field configurations with $N_{\rm f}
= 2+1$ dynamical quark flavors and $m_{\pi} \approx 200$ MeV and
$m_K\approx487$ MeV. Hermitian correlation matrices using both single baryon
and meson-baryon interpolating operators for a variety of different total
momenta and irreducible representations are used. Several parametrizations of
the two-channel scattering $K$-matrix are utilized to obtain the scattering
amplitudes from the finite-volume spectrum. The amplitudes, continued to the
complex energy plane, exhibit a virtual bound state below the $\pi\Sigma$
threshold and a resonance pole just below the $\bar{K}N$ threshold.
