note that 101 is odd.

therefore, det(A) = det(-A^{t}) = (-1)^{101}det(A^{t}) = -det(A) (since det(A) = det(A^{t})),

hence det(A) = 0.

now A^{1001}+ A^{1003}= (A^{1001})(I + A^{2}),

thus det(A^{1001}+ A^{1003}) = det(A^{1001})det(I + A^{2})

= (det(A))^{1001}(det(I + A^{2})) = (0)(det(I + A^{2})) = 0