Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 3 / e / Solution

Set $c=\cos (2{\alpha }_1)$ and $s=\sin (2{\alpha }_1)$. Multiplying the matrices, or using the Pauli matrix multiplication law, givesCollecting the coefficients of $i{\tau }_j$ in $M^{\prime }$ yieldsThis is the right-handed active rotation by $2{\alpha }_1$ about the positive $x_1$ axis. Likewise, with $c_j=\cos (2{\alpha }_j)$ and $s_j=\sin (2{\alpha }_j)$,Thus the other axes also have rotation angle $2{\alpha }_j$. The factor of two is essential: replacing $A_j$ by $-A_j$ leaves the conjugation unchanged, illustrating the two-to-one relation between SU(2) and three-dimensional rotations.