Determine Whether Aⁿ is Symmetric or Skew-Symmetric

Given:

\[ A \text{ is symmetric } \Rightarrow A^T = A \]

Step 1: Take Transpose of Aⁿ

\[ (A^n)^T = (A^T)^n \]

Step 2: Use Property

\[ (A^n)^T = A^n \]

Conclusion:

\[ (A^n)^T = A^n \Rightarrow A^n \text{ is symmetric} \]

Final Answer:

\[ \boxed{\text{A}^n \text{ is symmetric}} \]

Every power of a symmetric matrix remains symmetric.