1.6.3 Fields, Groups, Rings

Problem: On the set \( A=\left\{\langle a, b\rangle \mid a, b \in R, a^{2}+b^{2} \neq 0\right\} \) the operation \( \langle a, b\rangle *\langle c, d\rangle=\langle a c-b d, a d+b c\rangle \) is defined. Define the transition operation \( { }^{-1} \) to the inverse element so that the set \( \left\langle A, *,{ }^{-1}\right\rangle \) is a group.