Levi-Cevita symbols: Why is $\epsilon_{ijk}\epsilon_{pjk}$ equal to $2\delta_{ip}$, but not $0$?

Question

I'm learning vector calculus on my own and sometimes strange things happen that I don't know how I should explain them. We have this famous equality:

$$\epsilon_{ijk}\epsilon_{pqk}=\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp}$$

Now, if we set $j=q$ we get $$\epsilon_{ijk}\epsilon_{pjk}=\delta_{ip}\delta_{jj}-\delta_{ij}\delta_{jp}=\delta_{ip}-\delta_{ip}=0$$

But apparently the correct equality is $$\epsilon_{ijk}\epsilon_{pjk}=2\delta_{ip}$$

Why is it so? Where's my mistake? :|

Answer 1

In the second equation, you sum over $j$, so $\delta_{ip}\delta_{jj}$ should be evaluated as $3\delta_{ip}$.