Determinant Expansions: Intuitive Understanding & Proofs

[heshbon]

Hi, I can't understand why any determinant expansion is the same via any row/column.
My lecturer says the proof is too technical to go over,
Does anyone have a good way to think about it intuitavely or know a site which has a full proof?
thnx

 

[CompuChip]

This is a bit of a vague argument, but I think it is the underlying thought:
The determinant is not so much a property of a matrix, but actually a property of the map which that matrix represents (and usually as such can be assigned a geometric meaning). Therefore, it should not depend on details like which basis we happen to write the matrix in.

I'm not sure we can offer you much rigor without actually going into the proof

I think the proof is here by the way, at least for expansion along the first row and column. You can just do a basis transformation (permute the basis vectors) to get any row/column you want as the first one.