Undergrad Transpose Property (where's my mistake)

[member 428835]

Hi PF!

When proving $\left(AB\right)^T = B^T A^T$ I was thinking of writing $\left(AB\right)_{ij} = A_{ik} B_{kj} = D_{ij}$. Then $\left(D\right)^T_{ij} = D_{ji} = A_{jk} B_{ki} = A^TB^T$ but clearly this is incorrect. Can someone tell me where my mistake is made?

Thanks!

 

[blue_leaf77]

$\left(D\right)^T_{ij} = D_{ji} = A_{jk} B_{ki} = A^TB^T$ but clearly this is incorrect.

Yes, the last equality is incorrect.
Hint: $A_{jk}B_{ki} = (A^T)_{kj}(B^T)_{ik}$.

 

[micromass]

Hi PF!

When proving $\left(AB\right)^T = B^T A^T$ I was thinking of writing $\left(AB\right)_{ij} = A_{ik} B_{kj} = D_{ij}$. Then $\left(D\right)^T_{ij} = D_{ji} = A_{jk} B_{ki} = A^TB^T$ but clearly this is incorrect. Can someone tell me where my mistake is made?

Thanks!

You are using Einstein notation, right?

 

[member 428835]

You are using Einstein notation, right?

I am, and I think I found the mistake. $A$ initially had a different index, so changing that doesn't make sense, correct?