Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 19I / b / i / Solution

Direct matrix multiplication giveswhile every other pair among $A,B,C,D$ commutes. Also $Z=-I$ is central. Consequently every commutator of words in the generators lies in $⟨Z⟩$, so $G^{\prime }\subseteq ⟨Z⟩$. Since $Z=[A,B]$, the reverse inclusion holds and