I read a very slick proof of determinant properties, in this case of the fact $\det A = \det A^T$, which says in one place
It suffices to notice that for any elementary matrix $M$ we have $\det M = \det M^T$
But it's not obvious to me. How can we show it?
Note: we're using the Laplace expansion for rows as the definition. It was noted that for elementary matrix $M$ we have $\det M = 1$ if $M$ adds a multiplied row, $\det M = -1$ if $M$ swaps two rows and $\det M = c$ if $M$ multiplies one row by $c$.