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.

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- July 20th 2012, 11:09 AMloui1410A quick question in determinants
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. - July 20th 2012, 05:55 PMDevenoRe: A quick question in determinants
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