A technique for constructing the nucleon-nucleon transition matrices T in the presence of inelastic channels is given. The present technique avoids the usual potential fit and continues T off the energy shell directly consistent with the two-body scattering data and consistent with the properties of possible two-body bound states. The respective T matrices are determined by a real symmetric matrix function ⟨E1α1|σ|E2α2⟩ of the continuous energy variable E and the discrete eigenchannel label α. The on-shell part of σ is given by the eigenphases; the arbitrariness of T is the off-shell continuation of the phase-shift matrix ⟨Eα1|σ|Eα2⟩ into two dimensions. The present technique of constructing T makes the experimental nucleon-nucleon and meson-nucleon scattering data readily available as input for nuclear theory extended to include baryon resonances.