Question

What does the determinant represent in higher dimensions?

Answer 1

It measures the “space” contained by an n-dimensional parallelepiped. In 2d, it’s called “area” and in 3d, it’s called volume. In general, it is just “n-dimensional volume”. Technically, I think the absolute value of the determinant gives the volume, and the signed value tells you about its orientation in space (ie, if it’s “below” an odd number of axes, it will be negative) but I haven’t really thought about that in a while.

I don’t know about the application of “determinant as volume” specifically. The determinant itself has many applications in various fields. Most obvious example that I can come up with is that it used in linear algebra to determine if a system of n equations can be solved.

Answer 2

I don't know if I've ever run into an application of a geometric interpretation of the determinant as an area or volume. In the cases I'm familiar with, what's important is whether the determinant is positive, negative or zero. Or close to zero.