Question #41023

The trap tests the combinatorial counting of matrices over roots of unity under non-commutative constraints ($A\bar{A} \neq \bar{A}A$).

Students must equate matrix elements, leading to trigonometric conditions that partition the roots.

The 'Lethal Collision' occurs when counting the pairs $(a,d)$ satisfying $(a-d)/b \in \mathbb{R}$.

Students correctly identify the two cases ($a=d$ and $ad=-b^2$) but fail to subtract the overlap ($a=\pm ib$).

Because $i$ is a 12th root of unity, this overlap always exists and often manifests as real roots (e.g., when $b=i$, $a=\pm 1$).

Ignoring this overlap leads to double-counting and the trap answer 20160.