What is this matrix doing?

$\begin{aligned} \begin{bmatrix} A \\ B \\ C \end{bmatrix} & = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2 \\ 1+a^2 \\ 1+a \end{bmatrix} \\ & = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} + \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 \\ a^2 \\ a \end{bmatrix} \\ & = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 1 - a^2 \\ a^2 - a \\ a - 1 \end{bmatrix} \\ & = (a-1)\begin{bmatrix} a^2 \\ a \\ 1 \end{bmatrix} \end{aligned}$

Therefore,

$\begin{aligned} X & = |A+B+C| + |A|+|B|+|C| \\ & = |(a-1){\color{#3D99F6}(a^2+a+1)}| + |(a-1)a^2| + |(a-1)a| + |(a-1)| & \small \color{#3D99F6} \text{Note that }a^2 + a + 1 = 0 \\ & = 0 + |a-1|\color{#3D99F6} (|a^2|+|a|+1) & \small \color{#3D99F6} \text{and }|a^2| = |a| = 1 \\ & = 3 \left|{\color{#3D99F6}e^{\frac 23\pi i}} - 1\right| = 3 \left|{\color{#3D99F6}-\frac 12 + \frac {\sqrt 3}2 i} - 1\right| = 3 \left|-\frac 32 + \frac {\sqrt 3}2 i \right| & \small \color{#3D99F6} \text{Euler's formula: }e^{\theta i} = \cos \theta + i\sin \theta \\ & = 3\sqrt{\frac 94+\frac 34} = 3\sqrt 3 \approx \boxed{5.196} \end{aligned}$

After performing the matrix multiplication out, one obtains:

$A = 1 - a^2 = 1 - e^{j4\pi / 3} = \frac{3}{2} + \frac{\sqrt{3}}{2}j$ ;

$B = a^2 - a = e^{j4\pi / 3} - e^{j2\pi / 3} = -\sqrt{3}j;$

$C = a - 1 = e^{j2\pi /3} - 1 = -\frac{3}{2} + \frac{\sqrt{3}}{2}j$

and the following magnitudes:

$|A+B+C| = 0, |A| = |C| = \sqrt{(3/2)^2 + (\sqrt{3} / 2)^2} = \sqrt{3}, |B| = \sqrt{3}$

and the final sum:

$|A+B+C| + |A| + |B| + |C| = \boxed{3\sqrt{3}}.$