Question

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

Answer 1

If B is a symmetric matrix, then
\[B^T = B\]
`(AB   A^T)^T = (A^T) ^T B^T A^T    [∵ABC )^T = C^T B^T A^T] `
`( AB     A^T )^T = A B^T A^T  [∵ ( A^T)^T = A] `
`( AB     A^T )^T = AB   A^T  [∵B^T = B ] `

 

 `∴   AB  A^T` is a symmetric matrix .