Let AT = -A be a real, skew-symmetric n n matrix. (a) Prove that the only possible real eigenvalue of A is =

Let AT = -A be a real, skew-symmetric n × n matrix.
(a) Prove that the only possible real eigenvalue of A is λ = 0.
(b) More generally, prove that all eigenvalues λ of A are purely imaginary, i.e., Re λ = 0.
(c) Explain why 0 is an eigenvalue of A whenever n is odd.
(d) Explain why, if n = 3, the eigenvalues of A ‰  O are 0, i w, - i w, for some real w ‰  0.
(e) Verify these facts for the particular matrices

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