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Trying to understand the proof of:

$A^T*A = (A^T*A)^T$

The solution was given here

I thought one would end with $A_{ik}*A_{jk}$, can anyone explain the fault in my reasoning?

Thank you.

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$$ (A^T*A)_{ij}=(A^T)_{ik} A_{kj}=A_{ki}A_{kj};\\ $$ $$ \left(\,(A^T*A)^T\right)_{ij}=(A^T*A)_{ji}=(A^T)_{jk}A_{ki}=A_{kj}A_{ki} $$ equal to each other and to $ A_{ki}A_{kj}$.

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