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.