Question

If A is a skew-symmetric and n ∈ N such that (Aⁿ)^T = λAⁿ, write the value of λ.

Answer 1

\[\Rightarrow A^T = - A\]

\[\left( A^n \right)^T = \lambda A^n \] 

\[ \Rightarrow \left( A^T \right)^n = \lambda A^n \] 

\[ \Rightarrow \left( - A \right)^n = \lambda A^n \] 

\[ \Rightarrow \left( - 1 \right)^n A^n = \lambda A^n \] 

\[ \Rightarrow \lambda = \left( - 1 \right)^n\]