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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. The explicit description of the resolution by generators and relations is deferred to a later paper. 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.

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