5.1.4 Affine transformations

Problem: The improper motion is given by \[ X^{\prime}=\frac{1}{3}\left(\begin{array}{ccc} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{array}\right) X+\left(\begin{array}{l} 4 \\ 0 \\ 2 \end{array}\right) \] It is a composition of symmetry with respect to the plane \( 2 x_{1}-4 x_{2}+a x_{3}=6 \) and parallel translation, moreover \( a=\cdots \).