# Levi-Civita symbol and Kronecker delta

1. Feb 23, 2010

### typhoonss821

Hello everyone, I am stuck when I study Levi-Civita symbol.
The question is how to prove

$$\varepsilon_{ijk}\varepsilon_{lmn} = \det \begin{bmatrix} \delta_{il} & \delta_{im}& \delta_{in}\\ \delta_{jl} & \delta_{jm}& \delta_{jn}\\ \delta_{kl} & \delta_{km}& \delta_{kn}\\ \end{bmatrix}$$

where $$\varepsilon_{ijk}$$ represents Levi-Civita symbol and $$\delta_{il}$$ represents kronecker symbol.

Thank you very much^^

2. Feb 23, 2010

### slider142

Have you already established the identity $\epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}$?

3. Feb 23, 2010

### typhoonss821

Yes I have, but I don't know how to relate it to determinant.....

4. Feb 24, 2010

### Landau

Well, you could just write out that determinant and see what happens.