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