Question

If B is a skew-symmetric matrix, write whether the matrix AB A^T is symmetric or skew-symmetric.

Answer 1

If B is a skew-symmetric matrix, then

\[B^T = - B\]

\[\left( AB A^T \right)^T = \left( A^T \right)^T B^T A^T \left[ \because \left( ABC \right)^T = C^T B^T A^T \right]\] 

\[ \Rightarrow \left( AB A^T \right)^T = A B^T A^T \left[ \because \left( A^T \right)^T = A \right]\] 

\[ \Rightarrow \left( AB A^T \right)^T = A\left( - B \right) A^T \left[ \because B^T = - B \right]\] 

\[ \Rightarrow \left( AB A^T \right)^T = - AB A^T \] 

\[ \therefore AB A^T \text{is a skew - symmetric matrix} .\]