TITLE: What is the physical significance of the determinants of orthogonal matrices having the value of $\pm 1$?
QUESTION [13 upvotes]: I'm new to linear algebra and while studying orthogonal matrices, I found out that their determinant is always $\pm 1$. Why is that so? What could be the physical significance behind it?
I know that linear algebra can be intuitive when visualized, which 3B1B's videos made me realize, hence I would like to know more about this. Thanks in advance!

REPLY [5 votes]: Orthogonal matrices/transformations are essentially the mathematical way to speak about rotations (and/or reflections). This physical interpretation gives an easy way to see that they should have $\det(O)=\pm 1$.
The special thing about (certain) rotations (and/or reflections) in contrast to general linear transformations is, that by applying them multiple times, you come back to the identitiy transformation. E.g. turning to the left four times (by $45^\circ$) brings you back to your original orientation. Mathematically, this means $O^n=\mathrm{Id}$, where $\mathrm{Id}$ is the identity transformation that does nothing, and $n$ is the number of turns you have to make to be in the original orientation again. Now use some properties of the determinant:
$$\det(O)^n=\det(O^n)=\det(\mathrm{Id})=1.$$
This leaves us with no option besides $\det(O)$ being a root of unity, which can only be $\pm1$ in the real numbers.
The reasoning explained above only works for rotations with angles $2\pi/n$ for some $n\in\Bbb N$. However, it can be extended to rotations around arbitrary rational or irrational angles.