Find λ such that (Aⁿ)^T = λAⁿ

Given:

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

Step 1: Take Transpose of Power

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

Step 2: Substitute A^T = -A

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

\[ (A^n)^T = (-1)^n A^n \]

Conclusion:

\[ \lambda = (-1)^n \]

Final Answer:

\[ \boxed{\lambda = (-1)^n} \]

So, λ = 1 if n is even and λ = -1 if n is odd.