Solution
= Solution
For $A_1$, the <determinant> is $\cos^2\alpha_1-(i\sin\alpha_1)^2=1$. For $A_2$ it is $\cos^2\alpha_2+\sin^2\alpha_2=1$, and for $A_3$ it is $e^{i\alpha_3}e^{-i\alpha_3}=1$. Multiplicativity of the <determinant> gives
$$
\boxed{\det A_1=\det A_2=\det A_3=1,\qquad\det(A_1A_2A_3)=1.}
$$
Together with <unitarity>, this places the <matrices> and their product in <SU(2)>.