Levi-Civita help

Question

$\epsilon_{ijk}$ is the Levi-Civita tensor which is totally anti-symmetric. Let $A^{ijk}$ be a totally symmetric matrix. Is it true that $$\epsilon_{ijk}A^{ijk}=0?$$ I know this is the case for $\epsilon_{ij}A^{ij}=0$ in two dimensions. Also, do we know something about $$\epsilon_{ijk}A^{kjl}?$$

Answer 1

Yes. Because

$$I=\epsilon_{ijk}A^{ijk}=-\epsilon_{jik}A^{ijk}=-\epsilon_{jik}A^{jik}=-I$$.

So $I = 0$