Originally Posted by

**topspin1617**

Though I'm drawing a complete blank on how to compute the transpose using elementary matrices LOL.

Why does it matter? Once you've proven that $\displaystyle \det\left(E\right)=\det\left(E^{\top}\right)$ for all elementary matrices then we just see that

$\displaystyle \begin{aligned}\det\left(A^{\top}\right)&=\det\lef t(\left(E_1\cdots E_n\right)^{\top}\right)\\ &=\det\left(E_n^{\top}\cdots E_1^{\top}\right)\\ &=\det\left(E_n^{\top}\right)\cdots\det\left(E_1^{ \top}\right)\\ &=\det\left(E_n\right)\cdots\det\left(E_1\right )\\ &=\det\left(E_1\right)\cdots\det\left(E_n\right )\\ &=\det\left(E_1\cdots E_n\right)\\ &=\det\left(A\right)\end{aligned}$