TITLE: Use row reduction to prove that $\det(\mathbf{A})=\det(\mathbf{A}^{T})$
QUESTION [5 upvotes]: I need to prove that the determinant of a matrix is equal to the determinant of its transpose. This fact is obviously easy to prove using the definition of the determinant, but the question stipulates that the proof must be by row reduction.
It would never occur to me to do it this way. How do you even get to the transpose using row reduction? I've always just swapped the rows and columns. Any advice would be appreciated.

REPLY [4 votes]: Row-reducing $A$ to $R$ is done by multiplying $A$ by a sequence of elementary matrices $E_i$:
$$R = E_n E_{n-1}\ldots E_1 A$$
where each $E_i$ is one of three types of elementary matrices: row switching, row multiplication, or row addition (see http://en.wikipedia.org/wiki/Elementary_row_operations).
Then $R^T = A^T E_1^T \ldots E_n^T$ and
\begin{align*}
\det R &= \det E_1 \ldots \det E_n \det A\\
\det R^T = \det A^T \det E_1^T \ldots \det E_n^T &= \det E_1^T \ldots \det E_n^T \det A^T.
\end{align*}
Can you deduce the relationship between $\det R$ and $\det R^T$? What about between $\det E_i$ and $\det E_i^T$, for each of the three types of elementary matrices?