Proving Det(A) From Quantum Field Theory Problem

[Petraa]

Homework Statement

I was following this book "problem book in quantum field theory by voja radovanovic" and I got stuck in the following problem

Prove [itex]\epsilon_{\alpha\beta\gamma\delta}A_{\,\nu}^{\alpha}A_{\,\mu}^{\beta}A_{\,\lambda}^{\gamma}A_{\,\sigma}^{\delta}=\epsilon_{\mu\nu\lambda\sigma}\det\left(A\right)
[/itex]

Homework Equations

A is a matrix and in the left-hand side we have the components of this matrix

The Attempt at a Solution

I've tried to write an explicit form for the determinant using

[itex]
\det\left(A\right)=\sum_{i_{1}...i_{n}}\epsilon_{i_{1}}...\epsilon_{i_{n}}A_{1,i_{1}}...A_{n,i_{n}}[/itex]

but i didn't find anything useful. Any help or tip would be appreciated

Thank you

 

[mighty2000]

for reaching out for help with this problem. It seems like you are on the right track by trying to use the explicit form of the determinant. One tip I can offer is to try using index notation to simplify the expression and see if that leads you to a solution. Additionally, you may want to consider using properties of the Levi-Civita symbol to help simplify the expression. I hope this helps and good luck with your problem!