  In our paper "Non-commutative desingularization of determinantal varieties,
I" we constructed and studied non-commutative resolutions of determinantal
varieties defined by maximal minors. At the end of the introduction we asserted
that the results could be generalized to determinantal varieties defined by
non-maximal minors, at least in characteristic zero. In this paper we prove the
existence of non-commutative resolutions in the general case in a manner which
is still characteristic free, and carry out the explicit description by
generators and relations in characteristic zero. As an application of our
results we prove that there is a fully faithful embedding between the bounded
derived categories of the two canonical (commutative) resolutions of a
determinantal variety, confirming a well-known conjecture of Bondal and Orlov
in this special case.
