Expressing a general form as a sum of determinants

Let A = (a_ij) be a non-negative integer k×k matrix. A is a homogeneous matrix if a_ij + a_kl = a_il + a_kj for any choice of the four indexes. We ask: If A is a homogeneous matrix and if F is a form in C[x_1, . . . x_n] with deg(F) = trace(A), what is the least integer, s(A), so that F = det M_1 + · · · + det M_s(A), where the M_i = (F_ilm) are k × k matrices of forms and deg F_ilm = a_lm for every 1 ≤ i ≤ s(A)? We consider this problem for n ≥ 4 and we prove that s(A) ≤ k^{n−3} and s(A) < k^{n−3} in infinitely many cases. However s(A) = k^{n−3} when the integers in A are large with respect to k.