What is the intuition for why the determinant of a matrix is the product of the pivots in the matrix after gaussian elimination?

Question

I'm trying to connect the calculations for the determinant I'm seeing and the idea that the determinant is a signed measure of the factor by which the volume bounded by the basis vectors of a vector space changes when those basis basis vectors are acted on by a linear transformation. Does anyone have a good way of connecting this geometric intuition with the way matrix determinants are computed, specifically with the "product of pivots" method I refer to in the title to this post?

Thanks.

Answer 1

I can't give an explanation, because what you said is not true. Here is a counterexample:

$$ \det \left(\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix}\right)=0\neq1\times1.$$