Determine Whether An is Symmetric or Skew-Symmetric
Given:
\[ A \text{ is skew-symmetric } \Rightarrow A^T = -A \]
\[ n \text{ is even} \]
Step 1: Take Transpose
\[ (A^n)^T = (A^T)^n \]
Step 2: Substitute AT = -A
\[ (A^n)^T = (-A)^n = (-1)^n A^n \]
Step 3: Since n is Even
\[ (-1)^n = 1 \]
\[ (A^n)^T = A^n \]
Conclusion:
\[ A^n \text{ is symmetric} \]
Final Answer:
\[ \boxed{\text{A}^n \text{ is symmetric}} \]
Even powers of a skew-symmetric matrix are always symmetric.