Determinant of the transpose of elementary matrices

Is there a 'nice' proof to show that $\det(E^T) = \det(E)$ where $E$ is an elementary matrix?

Clearly it's true for the elementary matrix representing a row being multiplied by a constant, because then $E^T = E$ as it is diagonal.

I was thinking for the "row-addition" type, it's clearly true because if $E_1$ is a matrix representing row-addition then it is either an upper/lower triangular matrix and so $\det(E_1)$ is equal to the product of the diagonals. If $E_1$ is an upper/lower triangular matrix, then $E_1^T$ is a lower/upper triangular matrix and so $\det(E_1^T) = \det(E_1)$ as the diagonal entries remain the same when the matrix is transposed.

How about for the "row-switching" matrix where rows $i$ and $j$ have been swapped on the identity matrix? Can we use the linearity of the rows in a matrix somehow?

Thanks for any help!

• It is not true that $\det{M} = \mathrm{tr}\,M$ for upper or lower triangular matrices. – josh Mar 20 '13 at 1:35
• @josh Sorry, I've been massively stupid and thought the trace was the product of the diagonals NOT the sum. – Noble. Mar 20 '13 at 1:37

You can use the fact that switching two rows or columns of a matrix changes the sign of the determinant. Switching two rows of $E$ makes it diagonal, then switch the corresponding columns and you have $E^T$