 Solution

Multiplication by $A$ rotates the $(x,y)$ components by $\theta$ counterclockwise when viewed from positive $z$ toward the origin, leaving $z$ unchanged. It is a proper rotation about the $z$ axis, with determinant $+1$. Multiplication by $B$ sends $(x,y,z)$ to $(x,y,-z)$: it is a reflection in the $xy$ plane, with determinant $-1$. Neither operation changes a vector's Euclidean norm.