What does such a matrix do geometrically? I'm having a hard time visualizing this; I used some basic examples but I am not seeing a general pattern. Any help would be appreciated.
Edit: From the examples I've seen it looks kinda like a glide reflection, but again, I'm having a hard time coming up with the general case.
Well, it preserves length, so it's definitely a rigid motion. It obviously can't be a rotation since all rotations have determinant 1. A determinant of -1 means it reverses orientation, so it's gonna be a reflection, or a rotation and a reflection. I think it's the latter but I really don't know how to go about finding it and proving it.