Recall and Application Kronecker \(\delta\) and Levi-Civita \(\epsilon\) symbols

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  1. Writing out all terms for a \(2\times 2\) matrix \(A\), explicitly verify that \[ \epsilon_{ij} \epsilon_{mn} A_{im} A_{jn} = 2\det A. \] where the indices \(i,j,m,n\) take values 1 and 2.
  2. For a three by three matrix A show that \[ \det A = \epsilon_{ijk} A_{i1}A_{j2} A_{k3}.\]

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