Is there a rotation matrix that can invert any 3D vector?

Question

For example, could a single rotation matrix convert the following vectors:

vec{1, 1, 1} to vec{-1, -1, -1}

vec{1, -2, 4} to vec{-1, 2, -4}

Answer 1

Not even products of rotation matrices will be able to do that. In general, rotations preserve orientations (and so have determinant $1$), but in $\mathbb{R}^3$, the map you're describing has determinant $-1$, so no single rotation matrix nor product of them can reflect through the origin. You need to be able to reflect across some plane.