Let A be a real 101x101 matrix.
If A is anti-symmetric, then |A^1001 + A^1003|= 0.
Why is this true? This was the correct answer to a multiple-choice question.
note that 101 is odd.
therefore, det(A) = det(-At) = (-1)101det(At) = -det(A) (since det(A) = det(At)),
hence det(A) = 0.
now A1001 + A1003 = (A1001)(I + A2),
thus det(A1001 + A1003) = det(A1001)det(I + A2)
= (det(A))1001(det(I + A2)) = (0)(det(I + A2)) = 0