# Distributive property of a matrix on a cross product

Let $$\vec x, \vec y \in \mathbb{R}^3$$ and $$\bf A$$ be a $$3 \times 3$$ real matrix. Under what conditions does $$\bf A$$ distribute over a cross product:

$$\mathbf{A} (\vec x \times \vec y) = (\mathbf{A}\vec x) \times (\mathbf{A} \vec y)$$

The cross product in this case is the vanilla one over $$\mathbb{R}^3$$: $$(\vec a \times \vec b)_i = \epsilon_{ijk} a_j b_k$$

Where $$\epsilon$$ is the Levi-Civita symbol. My suspicion is matrices where $$\mathbf{A}^{-1} = \mathbf{A}^T$$ like the rotations in $$\mathbb{R}^3$$ would satisfy this condition. I can imagine the equality holding when $$\vec x$$ is aligned with the axis of rotation (i.e. $$\mathbf{A}\vec x = \vec x$$) and therefore rotating $$\vec y$$ also rotates $$\vec x \times \vec y$$.

Would anyone back the general case up with a proof for me?

• Of course immediately after posting I found this question here: math.stackexchange.com/questions/279173/… Should I delete this question or allow it to me marked duplicate? – cms Dec 13 '18 at 22:58
• Yes, absolutely, DELETE! – Cosmas Zachos Jan 4 at 2:22