Inner product of symmetric and anti-symmetric tensors

Question

I have the following question

"Let $A_{ij}$ be a symmetric tensor and let $B_{ij}$ be an antisymmetric tensor. Prove that the inner product of $A_{ij}$ and $B_{ij}$ is zero."

How would I go about doing this? I know that $A_{ij}=A_{ji}$ and $B_{ij}=-B_{ji}$ but I'm not too sure how this helps.

Any help is greatly appreciated, thank-you in advance.

Answer 1

Contracting these equations, $A_{ij}B_{ij}=-A_{ji}B_{ji}=-A_{ij}B_{ij}$, where the second $=$ sign relabels the indices. Hence $2A_{ij}B_{ij}=0$.