For any square matrix $A$, the matrix $A + A'$ is:

Taking the transpose: $(A + A')' = A' + (A')' = A' + A = A + A'$. Since the matrix equals its own transpose it is symmetric. By contrast, $A - A'$ is always skew-symmetric. Every square matrix can be uniquely written as a sum $\frac{1}{2}(A + A') + \frac{1}{2}(A - A')$ of a symmetric and a skew-symmetric matrix.