Question

Let R₀ denote the set of all non-zero real numbers and let A = R₀ × R₀. If '*' is a binary operation on A defined by

(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A

Find the invertible element in A ?

Answer 1

\[\text{ Let } \left(\text{ m, n }\right) \text{ be the inverse of } \left( a, b \right) \forall \left( a, b \right) \in A . \text{ Then }, \] 
\[\left( \text{a, b} \right) * \left( \text{ m, n } \right) = \left( 1, 1 \right)\] 
\[ \Rightarrow \left( \text{am, bn} \right) = \left( 1, 1 \right)\] 
\[ \Rightarrow \text{am = 1  &  bn }= 1\] 
\[ \Rightarrow m = \frac{1}{a}\text{ & } n = \frac{1}{b}\] 
\[\text{ Thus }, \left( \frac{1}{a}, \frac{1}{b} \right)\text{ is the inverse of } \left( a, b \right) \forall \left( a, b \right) \in A .\]