Matrix, orthogonal and cross product

Question

Let A be ortogonal $3\times3$ matrix. How do I show that $$A(v\times w)=(\det A)(Av\times Aw)$$

Here \times is crossproduct and v and w are $3$-dim vector.

Answer 1

Restate the claim as $A^T(Av\times Aw)=(\det A)(v\times w)$. The $i$th component of the left-hand side is$$A_{ij}^T\epsilon_{jkl}(Av)_k(Aw)_l=\epsilon_{jkl}A_{ji}A_{km}A_{ln}v_mw_n=(\det A)\epsilon_{imn}v_mv_n,$$which is the $i$th component of the right-hand side.