If A is a skew-sy

Let A is a skew – symmetric matrix, then

A^T = - A …(i)

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

(Aⁿ)^T = (A^T)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