Matrix and Its Transpose
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.
Okay, that makes sense, but apparently my approach is wrong since I can't use the fact that the determinant of the upper triangle matrix is the same as the determinant of the lower triangle matrix (haven't proven the fact that we can use expansion of minors using any row or column). I am stuck on how to go about this. A hint given was:
Suppose we do an elementary row operation on , obtaining . Then . What sort of column operation does do on ?
edit: Nevermind, I've figured it out. Thanks again!