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Levi-Civita symbol and Kronecker delta

  1. Feb 23, 2010 #1
    Hello everyone, I am stuck when I study Levi-Civita symbol.
    The question is how to prove

    [tex]\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}\\

    where [tex]\varepsilon_{ijk}[/tex] represents Levi-Civita symbol and [tex]\delta_{il}[/tex] represents kronecker symbol.

    Thank you very much^^
  2. jcsd
  3. Feb 23, 2010 #2
    Have you already established the identity [itex]\epsilon_{ijk}\epsilon_{ilm} = \delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}[/itex]?
  4. Feb 23, 2010 #3
    Yes I have, but I don't know how to relate it to determinant.....
  5. Feb 24, 2010 #4


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    Science Advisor

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