If A is a skew-symmetric matrix and n is an even natural number, write whether An is symmetric or skew-symmetric or neither of these 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 an even natural number, then
(An)T = An
[∵ (-1)2 = 1, (-1)4 =1,… (-1)n = 1]
So, An is a symmetric matrix