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प्रश्न
If B is a skew-symmetric matrix, write whether the matrix AB AT is symmetric or skew-symmetric.
उत्तर
If B is a skew-symmetric matrix, then
\[\left( AB A^T \right)^T = \left( A^T \right)^T B^T A^T \left[ \because \left( ABC \right)^T = C^T B^T A^T \right]\]
\[ \Rightarrow \left( AB A^T \right)^T = A B^T A^T \left[ \because \left( A^T \right)^T = A \right]\]
\[ \Rightarrow \left( AB A^T \right)^T = A\left( - B \right) A^T \left[ \because B^T = - B \right]\]
\[ \Rightarrow \left( AB A^T \right)^T = - AB A^T \]
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