A matrix which is both symmetric as well as skew-symmetric is a null matrix.

Explanation

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

`=> a_(ij) = a_(ji),` for all `i,j->(1)`

Here `a_(ij)` are elements of matrix with `i` and `j` representing rows and columns of matrix.

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

`=> a_(ij) = -a_(ji),` for all `i,j`

`=>-a_(ij) = a_(ji)->(2)`

From (1) and (2),

`a_(ij) = -a_(ij)`

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