Prove epsilon permutation and kronecker delta function relationship.

ε ijk εilm=δ jlδ km−δ klδ jm

Proof:-

Consider

ε ijk=|δi1 δi2 δi3δ j1 δ j2 δ j3δk 1 δk 2 δk 3

| Equation a

For [i j k] = [1 2 3]

ε 123=|δ 11 δ12 δ13δ21 δ22 δ23δ31 δ32 δ33

|ε 123=|1 0 0

0 1 00 0 1|=1

For [i j k] = [2 13]

ε 213=|δ21 δ22 δ23δ 11 δ12 δ13δ31 δ32 δ33

|=|0 1 01 0 00 0 1|=−1

For [i j k] = [1 13]

ε 113=|δ11 δ 12 δ 13δ11 δ 12 δ 13δ31 δ 32 δ 33

|=|1 0 01 0 00 0 1|=0

Hence for even combinations of i,j,k equation a gives 1, for odd combinations gives -1 and for repetitions gives 0.

Now consider two matrices A, B of same order.

From the properties of matrices we know that

Det(A).Det(B)=Det(AB)

ε ijk εilm=|{δi1 δi2 δi3δ j1 δ j2 δ j3δk 1 δk 2 δk 3

}{δi1 δi2 δi3δ j1 δ j2 δ j3δ k 1 δ k 2 δk 3

}|