Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 8B (Vectors and Matrices)

The determinant factors asThus for $\mu \ne \pm 1$, the kernel is $\{0\}$ and the inhomogeneous system has exactly one solution.

At $\mu =-1$, row reduction givesand the augmented matrix has the same rank two as $A$, so there are infinitely many solutions. At $\mu =1$,but the augmented matrix has rank three while $A$ has rank two, so there is no solution. Therefore the number of solutions is

Writing $n$ as a column vector, the reflection isso its reflection matrix is

Choose the $z$-axis along the line of intersection of the two planes. Their unit normals lie in the $xy$-plane; after choosing the $x$-axis, takewhere $\theta$ is the oriented angle between the planes. The two reflection matrices restrict toon the $xy$-plane and both fix the $z$-axis. Their product isThus the composition of two plane reflections is a rotation about the intersection line through angle $2\theta$.

A plane reflection has determinant $-1$, while a rotation matrix has determinant $+1$. The determinant of their product, in either order, is therefore $-1$. Since every rotation has determinant $+1$, such a product can never be another rotation.