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