How to Show Pij = Pji When Epsilon ijk Pkj = 0?

[latentcorpse]

if $\epsilon_{ijk} P_{kj}=0$ how do we show $P_{ij}=P_{ji}$?

 

[dx]

Expand out the LHS for each i.

 

[latentcorpse]

do you mean just substitute in numbers and see what happens because that works - i was just wondering if i could do it without explicitly using numbers (i.e. all of it in index notation).

 

[dx]

For i = j, clearly A_ij = A_ji, so you only need to check the cases in which i ≠ j.

For example, setting i = 1 in ε_ijkA_kj = 0, you get ε₁₂₃A₃₂ + ε₁₃₂A₂₃ = A₃₂ - A₂₃ = 0 ⇒ A₃₂ = A₂₃.

You just need to do the same with i = 2 and i = 3.