I am trying to prove det(A) = det($A^T$), starting with the idea that every square matrix is the product of elementary matrices.
Is this true, even for the non-invertible square matrices?
So, I'd like to start with the identity matrix and multiply it with elementary matrices until I get my desired matrix. Then use the fact that $(AB)^T$ = $B^T$$A^T$.
(The proof using the definition of determinant with the signum function and permutation stuff is a little difficult and not instructive at all, in my opinion.)
Thanks,
Best Answer
The elementary matrices are invertible, so any product of them is also invertible. However, invertible matrices are dense in all matrices, and determinant and transpose are continuous, so if you can prove that $\det(A) = \det(A^T)$ for invertible matrices, it follows that this is true for all matrices.