The mapping $\alpha$ of $\mathbb{R}^{3}$ into itself is a reflection in the plane $x_{2}=x_{3}$. Find the matrix $A$ of $\alpha$ with respect to any basis of your choice, which should be specified.

The mapping $\beta$ of $\mathbb{R}^{3}$ into itself is a rotation about the line $x_{1}=x_{2}=x_{3}$ through $2 \pi / 3$, followed by a dilatation by a factor of 2 . Find the matrix $B$ of $\beta$ with respect to a choice of basis that should again be specified.

Show explicitly that

$$B^{3}=8 A^{2}$$

and explain why this must hold, irrespective of your choices of bases.