No, a matrix cannot be both symmetric and skew-symmetric at the same time.

  A matrix is symmetric if it is equal to its transpose. That is, for a matrix A, if A = A^T, then it is symmetric. On the other hand, a matrix is skew-symmetric if the transpose of the matrix equals the negative of the matrix itself. That is, for a matrix A, if A^T = -A, then it is skew-symmetric.

  If a matrix is both symmetric and skew-symmetric, it would mean that A = A^T and A^T = -A. Combining these two conditions, we have A = -A, which implies that all elements of the matrix are equal to their negation. However, this is only possible if all the elements of the matrix are zero.

  Therefore, the only matrix that can be both symmetric and skew-symmetric is the zero matrix, where all elements are zero. Any other matrix cannot simultaneously satisfy both conditions.