A Matrix Problem

Algebra Level 3

Read the following statements carefully.

$(1$ ) The elements on the main diagonal of a symmetric matrix are all zero.

$(2$ ) The elements on the main diagonal of a skew-symmetric matrix are all zero.

$(3$ ) For any square matrix $A, \dfrac{1}{2}(A + A')$ is skew-symmetric.

$(4$ ) For any square matrix $A, \dfrac{1}{2}(A - A')$ is skew-symmetric.

Which of the following statements are correct?