Given that A is a symmetric matrix and n ∈ N, we want to determine the nature of An.
A matrix A is symmetric if AT = A, where T denotes the transpose of the matrix.
To solve this, consider the matrix power An. We need to check if this power remains symmetric.
We begin by establishing a property:
If A is symmetric, then:
By induction, An is symmetric for any natural number n.
Thus, the correct option is that An is a symmetric matrix.