  This work is entirely devoted to construct huge families of indecomposable
arithmetically Cohen-Macaulay (resp. Ulrich) sheaves E of arbitrary high rank
on a general standard (resp. linear) determinantal scheme X\subset \PP^n of
codimension c \ge 1, n-c \ge 1 and defined by the maximal minors of a t \times
(t+c-1) homogeneous matrix A. The sheaves E are constructed as iterated
extensions of sheaves of lower rank. As applications: (1) we prove that any
general standard determinantal scheme X\subset \PP^n is of wild representation
type provided the degrees of the entries of the matrix A satisfy some weak
numerical assumptions; and (2) we determine values of t, n and n-c for which a
linear standard determinantal scheme X\subset \PP^n is of wild representation
type with respect to the much more restrictive category of its indecomposable
Ulrich sheaves, i.e. X is of Ulrich wild representation type.
