A matrix ℳ represents a sequence of + and × operations. At the end you’ve linearly transformed a space (sheared it, expanded it, rotated it — but kept the origin where it is.)
Did the amount of stuff in the picture change when you did that? If you kept everything in proportion then det |ℳ| = 1. If not, then det |ℳ| ≠ 1.
If the amount of stuff increased by 10% then det |ℳ|=1.1. If you effectively shrank the picture in half, then det |ℳ|=.5. And so on.
The determinant |ℳ| is the change in volume after the linear transformation.
This metaphor extends to 3-D and beyond.
- If water is flowing linearly in a stream, then |ℳ| needs to be 1, or else water (matter) would be being created.
- If money is flowing linearly in a billion-dimensional economic system, then |ℳ| is hopefully just a little bit above 1, if value is being created. (Central banks need to print |ℳ| times more money to prevent deflation.)
- And a hundred-dimensional linear dynamical system’s phase space grows by |ℳ| at every step.
