# Proving a certain transposition

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.

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