solving the $ \varepsilon_{ijk}\varepsilon_{lmn}$(Levi Civita)

Question

How can I solve this : $ \varepsilon_{ijk}\varepsilon_{lmn}=??$ I know that It can be solve with 2 determinants but I don't know how.and I don't what are the determinants!

Answer 1

I assume that $\varepsilon_{ijk}$ is the Levi-Civita-symbol. The tricky think is that in this notation you actually already use Einsteins sum convention. In fact you can solve it by a single determinant.

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