Using row reduction, prove that the determinant of a matrix and the determinant of the matrix's transpose are equal.

This is my attempt, but I am not sure if this is actually using row reduction, and if this is the right approach, I am not totally sure on the details:

Suppose is a square matrix and suppose is a sequence of elementary matrices such that is an upper triangle matrix (I am not sure how to prove that such a sequence must always exist). Then clearly since the determinant of an upper triangle matrix and the determinant of a lower triangle matrix is the product of the entries along the main diagonal, which are the same in and .

Now this is the part where I am stuck. So I know that , but I don't know how to show/prove that for any elementary matrix , .

Any help would be appreciated it, thank you.