If A is a skew-symmetric matrix and n is an odd natural number, write whether An is symmetric or skew-symmetric or neither of the two.

Let A is a skew – symmetric matrix, then

AT = - A …(i)


Now, we have to check An is symmetric or skew – symmetric


(An)T = (AT)n [for all n Є N]


(An)T = (- A)n [from (i)]


(An)T = (-1)n (A)n


Given that n is odd natural number, then


(An)T = - An


[ (-1)3 = - 1, (-1)5 = - 1,… (-1)n = - 1]


So, An is a skew - symmetric matrix


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