  In this paper we address a conjecture by Kleppe and Mir\'o-Roig stating that
suitable twists by line bundles (on the smooth locus) of the exterior powers of
the normal sheaf of a standard determinantal locus are arithmetically
Cohen--Macaulay, and even Ulrich when the locus is linear determinantal. We do
so by providing a very simple locally free resolution of such sheaves obtained
through the so-called Weyman's Geometric Method.
