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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.
Options:
1. \(A^{2 n}\) is Skew-symmetric matrix.
2. \(A^{2 n+1}\) is Skew-symmetric matrix.
Solution:
1522 Upvotes
Verified Answer
The correct answer is:
1 is false, 2 is true
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.
\(\begin{aligned}
\therefore \quad A^T & =-A \Rightarrow\left(A^{2 n}\right)^T=\left(A^T\right)^{2 n}=(-A)^{2 n} \\
\left(A^{2 n}\right)^T & =A^{2 n}
\end{aligned}\)
\(\therefore A^{2 n}\) 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.
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