det(A+B)

def nb_changes(a,b):
  s = 1
  for x in a:
    for y in b:
      if x > y:
        s *= -1
  return s

def helper(A, B, as1, as2, bs1, bs2):
  M = A[as1,as2]
  N = B[bs1,bs2]
  return nb_changes(as1,bs1) * nb_changes(as2,bs2) * det(M) * det(N)

def det_sum(A,B,n):
  rn = Set(range(n))
  result = 0
  for ss1,ss2 in cartesian_product([Subsets(rn),Subsets(rn)]):
    if ss1.cardinality() != ss2.cardinality():
      continue
    Css1 = rn.difference(ss1)
    Css2 = rn.difference(ss2)
    r1 = result
    result += helper(A,B,list(ss1),list(ss2),list(Css1),list(Css2))
  return result

n = 7
A = random_matrix(ZZ,n,n)
B = random_matrix(ZZ,n,n)
print det_sum(A,B,n)
print det(A+B)