Can One Factor the Classical Adjoint of a Generic Matrix?

Let k be an integral domain, n a positive integer, X a generic n×n matrix over k (i.e., the matrix (xij) over a polynomial ring k[xij ] in n 2 indeterminates xij), and adj(X) its classical adjoint. For char k = 0 it is shown that if n is odd, adj(X) is not the product of two noninvertible n×n matrices over k[xij ], while for n even, only one particular sort of factorization can occur. Whether the corresponding result holds in positive characteristic is open. The operation adj on matrices arises from the (n−1)st exterior power functor on modules; the analogous factorization question is raised for matrix constructions arising from other functors.