Levi-Civita tensor contraction contradiction

Question

It's quite a time i am using the Levi Civita tensor for the efficency is index notation. But, if we use the one-common-index contraction when we have two common indices we get

$$ \epsilon_{ijk}\epsilon_{ijn} = \delta _{jj} \delta_{kn}-\delta_{jn}\delta_{kj} = \delta_{kn} - \delta_{kn} = 0$$ when we should get $2 \delta_{kn}$. Someone can please explain where i'm wrong?

Answer 1

As long as dimension is 3, $\delta_{jj} = 3$. Hence $$ \delta_{jj} \delta_{kn} - \delta_{jn}\delta_{kj} = 3\delta_{kn} - \delta_{kn} = 2\delta_{kn}.$$