If A is a skew-symmetric and n ∈ N such that
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.

$\Rightarrow A^{T}=-A$

$\left(A^{n}\right)^{T}=\lambda A^{n}$

$\Rightarrow\left(A^{T}\right)^{n}=\lambda A^{n}$

$\Rightarrow(-A)^{n}=\lambda A^{n}$

$\Rightarrow(-1)^{n} A^{n}=\lambda A^{n}$

$\Rightarrow \lambda=(-1)^{n}$