Proving det(w_ij) = det(u_ik v_kj) = det(v_kj u_ik) in Determinate Proof

[deserteagle778]

How to prove det(w_ij) = det(u_ik v_kj) =det(v_kj u_ik )

All I know so far is det(w_ij) = e_ijk w_1i w_2j w_3k. I have no idea what to do next.

 

[adjacent]

Welcome to PF
Use LaTeX,Your equations will look cleaner.
Found at the top right of the Advanced editor,(The sigma symbol)

 

[Office_Shredder]

So your goal is to show that if U and V are 3x3 matrices specifically, that det(UV) = det(VU)?

Surely from what you have written there is an obvious first step (what is each w equal to?) Have you tried that at all?

 

[deserteagle778]

I'm not quite sure what you mean Shredder. Yes U and V are 2nd order rank tensor 3x3.

 

[Office_Shredder]

You said that all you know is that det(w_ij) = e_ijk w_1i w_2j w_3k. But in your problem you are not working with an arbitrary matrix w_ij, you are working with a product UV and VU. The obvious step is to take your definition of the determinant and plug in UV and VU into it to write out things in terms of u_ij and v_ij.

This is a determinant of a 3x3 matrix, so there are only six terms. If you write it out for UV and VU and expand everything out you can easily check by hand that the two sides are equal to each other without needing to know anything about the determinant and matrix multiplication other than their definitions.

 

[deserteagle778]

Thanks

 

[Ray Vickson]

How to prove det(w_ij) = det(u_ik v_kj) =det(v_kj u_ik )

All I know so far is det(w_ij) = e_ijk w_1i w_2j w_3k. I have no idea what to do next.

Used standard properties of determinants; see, eg., http://aleph0.clarku.edu/~djoyce/ma130/determinants3.pdf , especially Theorem 4 on page 2.