Strange matter is believed to exist in the cores of neutron stars based on simple kinematics. If this is true, then hyperon-nucleon interactions will play a significant part in the neutron star equation of state (EOS). Yet, compared to other elastic scattering processes, there is very little data on $Lambda$-$N$ scattering. This experiment utilized the CLAS detector to study the $Lambda p rightarrow Lambda p$ elastic scattering cross section in the incident $Lambda$ momentum range 0.9-2.0 GeV/c. This is the first data on this reaction in several decades. The new cross sections have significantly better accuracy and precision than the existing world data, and the techniques developed here can also be used in future experiments.