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

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if \epsilon_{ijk} P_{kj}=0 how do we show P_{ij}=P_{ji}?
 
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Expand out the LHS for each i.
 


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).
 


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

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

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

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