What's the geometric meaning of a negative determinant?

Question

Geometrically, the determinant of a matrix is the signed volume of a unit cube after the transformation defined by the matrix is applied. However, I'm have trouble understanding what the "signed" mean here. Since effectively volumes are only positive (or zero but for now let's not worry about it). So what's the geometric meaning of a negative determinant? How should I understand the negative volume produced by applying such transformation?

Answer 1

The idea has to do with orientation, that is, which direction is right and which is left. Ultimately this has to do with which order is given for the basis elements. For example consider $\Bbb R^2$ with basis elements $(1,0)$ and $(0,1)$. We can consider $(1,0)$ to be pointing "to the right", and $(0,1)$ pointing " forward".

Putting $(1,0)$ and $(0,1)$ in a matrix gives the identity matrix whose determinant is $1$ of course.

Now switch the two basis elements. Imagine rotating the plane around the line $y=x$. Now $(1,0)$ is pointing forward but $(0,1)$ is pointing to the left.

Of course if we switch rows in the identity matrix the resulting matrix has determinant $-1$.