Question

Let A = R₀ × R, where R₀ denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows (a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R₀ × R :

Find the invertible elements in A ?

Answer 1

\[ \text{Let} F = (m, n) \text{be the inverse in A} \forall m \in R_0 \text{ & }n \in R\] 
\[X \odot F = E \text{ and } F \odot X = E\] 
\[ \Rightarrow \left( am, bm + n \right) = \left( 1, 0 \right) \text{ and } \left( ma, na + b \right) = \left( 1, 0 \right)\] 
\[\text{ Considering } \left( am, bm + n \right) = \left( 1, 0 \right)\] 
\[ \Rightarrow am = 1\] 
\[ \Rightarrow m = \frac{1}{a}\] 
\[\text{ & }bm + n = 0\] 
\[ \Rightarrow n = \frac{- b}{a} \left[ \because m = \frac{1}{a} \right]\] 
\[\text{ Considering } \left( ma, na + b \right) = \left( 1, 0 \right)\] 
\[ \Rightarrow ma = 1\] 
\[ \Rightarrow m = \frac{1}{a}\]

 \[\text{ & } na + b = 0\] 
\[ \Rightarrow n = \frac{- b}{a}\] 
\[ \therefore \text{ The inverse of } \left( a, b \right) \in \text{A with respect to} \odot \text{is} \left( \frac{1}{a}, \frac{- b}{a} \right) . \]