If A is a symmetric matrix and n ∈ N, write whether An is symmetric or skew-symmetric or neither of these two.
Given that A is a symmetric matrix
∴ A = AT …(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 = (AT × AT … AT)
[∵ (AB)T = BTAT]
= 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