We proofed in class that for any matrix $\det(A) = \det(A^T)$.
I was asked to prove the same, only for elementary matrices. Though repeating the proof for any matrix would do the work, it's like using heavy machinery and I guess it's not the intention of this exercise.
There are three row operations.
For the operation $R_i \rightarrow \alpha\cdot R_i$. We know that $I=I^T$. And so, $E = E^T$. Hence, $\det E = \det E^T$.
For the opertaion, $R_i \leftrightarrow R_j$. I've noticed a symmetry which sustained after transposing. I guess that's the key for this operation and for the third one.
I'm having trouble with formulating it into a rigorous proof.