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Question
If A is a skew-symmetric matrix and n is an odd natural number, write whether An is symmetric or skew-symmetric or neither of the two.
Solution
`If A is a skew - symmetric matrix, then A^T = - A`
`( A^n )^T = ( A^T)^n [ " For "all n ∈ N ]`
\[ \Rightarrow \left( A^n \right)^T = \left( - A \right)^n \left[ \because A^T = - A \right]\]
\[ \Rightarrow \left( A^n \right)^T = \left( - 1 \right)^n A^n \]
Hence, `( A)^n `is skew-symmetric when n is an odd natural number.
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