Show that positive odd integral powers of a skew-symmetric matrix are skew-symmetric and positive even integral powers of a skew-symmetric matrix are symmetric.

Explanation

Let `A` be a skew-symmetric matrix. Therefore `Aprime=-A`.


We know that `(A^{n})^{prime}=(A^{prime})^{n}` for all `n in N`.



`therefore quad(A^{n})^{prime} =(-A)^{n}`


`=(-1)^{n} A^{n}`


`therefore quadA^{n}prime =-A^{n}, quad text { if } n text { is odd. } `


`=A^{n}, quad text { if } n text { is even. }`



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