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Question
If A is a skew-symmetric and n ∈ N such that (An)T = λAn, write the value of λ.
Solution
\[\left( A^n \right)^T = \lambda A^n \]
\[ \Rightarrow \left( A^T \right)^n = \lambda A^n \]
\[ \Rightarrow \left( - A \right)^n = \lambda A^n \]
\[ \Rightarrow \left( - 1 \right)^n A^n = \lambda A^n \]
\[ \Rightarrow \lambda = \left( - 1 \right)^n\]
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