Does $\widehat{M}\ (\vec{a}\times\vec{b})=\widehat{M}\ \vec{a}\times\widehat{M}\ \vec{b}$?

Question

Is it true to write that $\widehat{M}\ (\vec{a}\times\vec{b})=\widehat{M}\ \vec{a}\times\widehat{M}\ \vec{b}$? If so, how can this be proven?

Answer 1

This is not true: consider $M = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}$, $\vec{a} = \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix}$, $\vec{b} = \begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix}$.