Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 1 / 18 / d

An orthogonal matrix has orthonormal columns. The first column of $L$ has squared norm $1+c^2$, so $c=0$. Orthogonality of the two columns then gives $-b+cd=-b=0$, hence $b=0$. The second column must have squared norm $d^2=1$. ThereforeBoth resulting matrices satisfy $L^{T}L=I$ directly.

Among the two orthogonal matrices from part (d)(i), $d=-1$ gives $L=-I$, a rotation matrix through angle $\pi$. For $d=1$, the determinant is $-1$, so the matrix reverses orientation and is a reflection rather than a rotation:It is the reflection matrix across the $y$ axis.