If a symmetric/skew symmetric matrix is expressed as a sum of symmetric and skew symmetric matrix then prove that one of the matrices in the sum must be zero.

$A$ and $B$ are square matrices of same order, prove that:

If $A, B \text{and} AB$ are all symmetric, then $AB-BA=0$.

$A$ and $B$ are square matrices of same order, prove that:

If $A, B \text{and} AB$ are all skew-symmetric, then $AB+BA=0$.

Express as a sum of a symmetric and a skew symmetric matrix

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 0& 1\\ -1& 5& -2\end{array}\right]$