Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 17 / c / ii / Solution

The matrix trace and determinant of the matrices in part (i) areSince $\mathrm{Tr}X=0$ and $\det X=2$, the extra conditions force $\alpha =0$ and ${\beta }^2=1$. HenceBoth choices are real, commute with $X$, and satisfy the required trace and determinant conditions.