Question:
If $A$ is a skew-symmetric and $n \in N$ such that $\left(A^{n}\right)^{T}=\lambda A^{n}$, write the value of $\lambda$.
Solution:
Given: A is skew symmetric matrix.
$\Rightarrow A^{T}=-A$
$\left(A^{n}\right)^{T}=\lambda A^{n}$
$\Rightarrow\left(A^{T}\right)^{n}=\lambda A^{n}$
$\Rightarrow(-A)^{n}=\lambda A^{n}$
$\Rightarrow(-1)^{n} A^{n}=\lambda A^{n}$
$\Rightarrow \lambda=(-1)^{n}$
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