If A is a symmetric matrix and n ∈ N, write whether An is symmetric or skew-symmetric or neither of these two.

If $A$ is a symmetric matrix, then $A^{T}=A$.

Now,

$\left(A^{n}\right)^{T}=\left(A^{T}\right)^{n} \quad[$ for all $n \in N]$

$\Rightarrow\left(A^{n}\right)^{T}=(A)^{n} \quad\left[\because A^{T}=A\right]$

Hence, $A^{n}$ is a symmetric matrix.