If A is a symmetr

Given that A is a symmetric matrix

∴ A = A^T …(i)

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

(Aⁿ)^T = (A×A×A×A…A)^T [for all n Є N]

⇒ (Aⁿ)^T = (A^T × A^T … A^T)

[∵ (AB)^T = B^TA^T]

= A × A … A [from (i)]

= Aⁿ

⇒ (Aⁿ)^T = Aⁿ

Case 1: If n is an even natural number, then

(Aⁿ)^T = Aⁿ

So, Aⁿ is a symmetric matrix

Case 2: If n is odd natural number, then

(Aⁿ)^T = Aⁿ

So, Aⁿ is a symmetric matrix