All positive integral powers of a symmetric matrix are :

Symmetric

Let $A$ be a symmetric matrix, i.e., $A^T = A$

Check $(A^n)^T$ for positive integer $n$:

$(A^n)^T = (A \cdot A \cdot ... \cdot A)^T = A^T \cdot A^T \cdot ... \cdot A^T = A \cdot A \cdot ... \cdot A = A^n$

Thus, $A^n$ is symmetric for all positive integers $n$.

Answer: Symmetric