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Show that the matrix B^TA B is symmetric...

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

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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`
...
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  • If A is symmetric as well as skew-symmetric matrix, then A is

    A
    diagonal matrix
    B
    null matrix
    C
    triangular materix
    D
    none of these

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