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On the lecture notes I found online there is the following proof that I don't understand.

Let A, B be two matrices.

$$(AB)^T=B^TA^T$$

I know this identity well, but I don't get this proof.

Here it goes: Let $C=AB$

$$C_{ij}^T=C_{ji}=A_{jk}B_{ki}=A_{kj}^TB_{ik}^T$$

And last step:

$$A_{kj}^TB_{ik}^T=B_{ik}^TA_{kj}^T$$

I clearly don't get something here since it seems that by ignoring the last step it was proven that $(AB)^T=A^TB^T$.

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The formula for the product $C=A^TB^T$ is $C_{ij}=A^T_{ik}B^T_{kj}$. Note, that the second index of $A^T_{ik}$ is equal to the first index of $B^T{kj}$. In your formula this indexes are different, by they will become equal when you change $A$ and $B$. This is why $B^TA^T$ not $A^TB^T$

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