  Let X be a standard determinantal scheme X \subset \PP^n of codimension c,
i.e. a scheme defined by the maximal minors of a t \times (t+c-1) homogeneous
polynomial matrix A. In this paper, we study the main features of its normal
sheaf \shN_X. We prove that under some mild restrictions: (1) there exists a
line bundle \shL on X \setminus Sing(X) such that \shN_X \otimes \shL is
arithmetically Cohen-Macaulay and, even more, it is Ulrich whenever the entries
of A are linear forms, (2) \shN_X is simple (hence, indecomposable) and,
finally, (3) \shN_X is \mu-(semi)stable provided the entries of A are linear
forms.
