Show that the matrix `B^TA B` is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.

Explanation

A matrix `A` is symmetric if `A = A^T`.

A matrix `A` is skew-symmetric if `A = -A^T`.

(i) When `A` is symmetric matrix, then `A = A^T`

Then, `(B^TAB)^T = B^TA^T(B^T)^T = B^TAB`...[As `A^T = A and (B^T)^T = B`]

`:. (B^TAB)^T = B^TAB`

`:. B^TAB` is a symmetric matrix when `A` is symmetric



(ii)When `A` is skew-symmetric matrix, then `A = -A^T`

...