Show that the symmetric and antisymmetric parts of a second-rank tensor are themselves tensors, and that the decomposition of a tensor into symmetric and antisymmetric parts is unique.

For the tensor $A$ having components

$$A=\left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 1 & 2 & 3 \end{array}\right)$$

find the scalar $a$, vector $\mathbf{p}$ and symmetric traceless tensor $B$ such that

$$A \mathbf{x}=a \mathbf{x}+\mathbf{p} \wedge \mathbf{x}+B \mathbf{x}$$

for every vector $\mathbf{x}$.