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Question
If A is a skew-symmetric matrix and n is an even natural number, write whether An is symmetric or skew symmetric or neither of these two.
Solution
`If A is a skew - symmetric matrix, then A^T = - A .`
\[ \left( A^n \right)^T = \left( A^T \right)^n \left[ \text{For all n }\in N \right]\]
\[ \Rightarrow \left( A^n \right)^T = \left( - A \right)^n \left[ \because A^T = - A \right]\]
\[ \Rightarrow \left( A^n \right)^T = \left( - 1 \right)^n A^n \]
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