You could have invented the determinant
I'm going to talk about how a geometric intuition becomes the definition of the determinant. As many of you already know, the determinant of a matrix is the (hyper)volume of the (hyper)parallelepiped whose edges are the (row or column) vectors of the matrix. This page on Wolfram gives you an interactive visualization of a parallelogram or a parallelepiped while showing the corresponding determinant. There are abundant sources on the internet (many on math.stackexchange.com ) that give more-or-less the same explanation as I just said above, but most of them don't explain why. That's what I want to discuss in this post. First goal: Define area in 2D I'll motivate this discussion with a simple question: how to define volume in a Euclidean space? I'm kidding. That question is actually not that simple, and people have gone to great lengths to answer that. Let me simplify it a bit more, quite gratuitously, so that we ...