python3 - <<'EOF'
import random
def det(A):
 n=len(A)
 A=[list(r) for r in A]
 for c in range(n):
  p=next((r for r in range(c,n) if A[r][c]),None)
  if p is None:return 0
  A[c],A[p]=A[p],A[c]
  for r in range(c+1,n):
   if A[r][c]:A[r]=[x^y for x,y in zip(A[r],A[c])]
 return 1
def invdiag(C):
 n=len(C)
 out=[]
 for i in range(n):
  rows=C[:i]+C[i+1:]
  sub=[row[:i]+row[i+1:] for row in rows]
  out.append(det(sub))
 return out
cf=0
for trial in range(2000):
 M=[[0]*6 for _ in range(6)]
 for i in range(6):
  M[i][i]=random.randrange(2)
  for j in range(i+1,6):
   M[i][j]=M[j][i]=random.randrange(2)
 if not any(M[i][i] for i in range(6)):continue
 if not det(M):continue
 v=[M[i][i] for i in range(6)]
 p=sum(v)%2
 x=1^p
 one=[sum(M[i][j] for j in range(6))%2 for i in range(6)]
 w=[x*v[i]^one[i] for i in range(6)]
 C=[[0]*8 for _ in range(8)]
 C[0][0]=C[1][1]=1
 C[0][1]=C[1][0]=p
 for i in range(6):
  C[0][i+2]=C[i+2][0]=v[i]
  C[1][i+2]=C[i+2][1]=w[i]
  for j in range(6):
   C[i+2][j+2]=M[i][j]^(v[i]&v[j])
 if not det(C):cf+=1
 if invdiag(C)!=[1]*8:cf+=1
print('fail',cf)
EOF