Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2019 / Paper 1 / 18 / a / ii / Solution

For either $A$ from part (i), chooseIndeed, using $A^2=-I$ and $A^{\mathsf{T}}=-A$,Its matrix trace is $-2/\sqrt{2}=-\sqrt{2}<0$. Explicitly, for the two $A$ matrices in the order displayed above, the corresponding orthogonal matrices areTheir negatives have positive matrix trace and would not meet the additional sign condition.