What can be said about the nature of rotations in terms of invertibility?

Rotations in the context of linear algebra are represented by rotation matrices, which are orthogonal matrices with a determinant of 1. One of the key characteristics of invertible matrices is that they must have a non-zero determinant. Since rotation matrices meet this criterion, they are always invertible.

When a rotation is applied to a vector in a vector space, it transforms that vector without changing its length, and the operation can be reversed. This means that for every rotation, there exists an inverse operation, which is the rotation by the negative of the angle used in the original rotation. Thus, for any angle, you can always find a corresponding rotation matrix that will take you back to the initial position of the vector.

In summary, the consistent nature of rotation matrices being orthogonal and having non-zero determinants confirms that they are always invertible.