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

Since $A^2=-I$, the matrix inverse is $A^{-1}=-A$. The orthogonal matrix condition also gives $A^{-1}=A^{\mathsf{T}}$, so $A^{\mathsf{T}}=-A$. Thus $A$ is skew-symmetric, forcing $a=0$ and $c=-b$. Its columns must be unit vectors, giving $b^2=1$. Therefore the two possibilities areBoth satisfy the original conditions by direct multiplication. They are the two orthogonal skew-symmetric matrices in two dimensions, representing opposite quarter-turn rotations.

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.