If $A$ is a Skew-symmetric matrix then (given $n \in \mathbf{N}$) 1. $A^{2 n}$ is Skew-symmetric matrix. 2. $A^{2 n+1}$ is Skew-symmetric matrix.

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Aicharya · Accepted Answer

Given, $A$ is a skew symmetric matrix $\begin{aligned} \therefore \quad A^T & =-A \Rightarrow\left(A^{2 n}\right)^T=\left(A^T\right)^{2 n}=(-A)^{2 n} \ \left(A^{2 n}\right)^T & =A^{2 n} \end{aligned}$ $\therefore A^{2 n}$ is Symmetric Matrix $\begin{aligned} & \left(A^{2 n+1}\right)^T=\left(A^T\right)^{2 n+1}=(-A)^{2 n+1} \ & \left(A^{2 n+1}\right)^T=-A^{2 n+1} \end{aligned}$ $A^{2 n+1}$ is skew symmetric Hence, option (d) is correct.

If \(A\) is a Skew-symmetric matrix then (given \(n \in \mathbf{N}\)) 1. \(A^{2 n}\) is Skew-symmetric…

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