A square matrix which is both symmetric and skew-symmetric is

The correct answer is Option (3) → a zero matrix

Let A be the matrix. Since A is symmetric and skew-symmetric,

    \(A' = A\)
    \(A' = -A\)

So

    \(A = -A\)
    \(2A = O\)
    \(A = O\)

Hence the matrix must be the zero matrix.