system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: what exactly is the determinant of a matrix calculating, and why does it end up being so significant/useful in other matrix calculations?
gpt
I'll focus on 3 x 3 matrices.

Take any 3D shape. Something that's asymetrical, like a sculpture of Vincent Van Gogh's head. Or anything else, it doesn't matter, but it should have a finite volume V. (and not zero volume, like a flat plane). Think of your 3D shape as sitting in an x,y,z coordinate system. There are infinitely many points in your shape and each of those points P is described by a vector that goes from (0,0,0) to P. So you have infinitely many vectors that describe your shape. 

Now suppose we have a matrix M. We can think of M as transforming vectors. Take a vector v and M\*v is some other vector (or maybe the same vector!).

We're going to take every vector that's describing your shape and transform each of them with M. The end result will be another shape. How different it is will depend a lot on M, but it can be any combination of the following: your original shape rotated about some axis; your original shape scaled up or down in size; a mirror image of your original shape; your original shape skewed, as if it were made out of clay and your smushed/stretched it uniformly along some combination of directions.

The volume of your transformed shape is likely different than it was originally. The factor by which the volume changed is the determinant of M. That is, 

New Volume = |det A| \* Old Volume.

I used absolute value on the determinant there because volume ought to be positive and the determinant can be negative. A negative determinant has a straightforward interpretation: it means the transformation of your shape involved taking a mirror image. Like Vincent van Gogh's missing ear switching sides.

With this volume-based interpretation of determinants we can quickly make some inferences about determinants that are pretty hard to perceive just from the definition of determinants. For one, a zero determinant must mean the transformation squished your shape to something with zero volume -- a flat plane, or a line, or a point. So any matrix that "projects" down to a plane or a line or a point must have zero determinant. Also we can infer if the transformation by M only rotated your shape without altering it otherwise, then the determinant of M should be 1, since the volume is the same. Likewise, a purely mirror-image transformation, or a mirror-image combined with a rotation, ought to have determinant -1. Also we can infer the important rule for determinants, that if K and M are both matrices, then det(K\*M) = det(K)\*det(M). I'll leave that for you to reason through.