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Question
Let A = Q ✕ Q, where Q is the set of all rational numbers, and * be a binary operation defined on A by (a, b) * (c, d) = (ac, b + ad), for all (a, b) (c, d) ∈ A.
Find
(i) the identity element in A
(ii) the invertible element of A.
(iii)and hence write the inverse of elements (5, 3) and (1/2,4)
Solution
(i)
Let E=(x, y) be the identity element in A with respect to *,∀ x, y∈Q such that X*E=X=E*X,∀ X∈A
⇒X*E=X and E*X=X
⇒(ax, b+ay)=(a, b) and (xa, y+xb)=(a, b)
Considering (ax, b+ay)=(a, b)
⇒ax=a
⇒x=1 and b+ay=b
⇒y=0 [∵x=1]
Also,
Considering (xa, y+xb(a, b)
⇒xa=a
⇒x=1and y+xb=b
⇒y=0 [∵ x=1]∴ (1, 0) is the identity element in A with respect to *.
(ii)
Let F=(m, n) be the inverse in A,∀ m, n∈Q
X*F=E and F*X=E
⇒(am, b+an)=(1, 0) and (ma, n+mb)=(1, 0)
Considering (am, b+an)=(1, 0)
⇒am=1
⇒m=1/a and b+an=0
⇒n=−b/a
Also,
Considering (ma, n+mb)=(1, 0)
⇒ma=1
⇒m=1/a and n+mb=0
⇒n=−b/a (∵m=1/a)
∴The inverse of (a, b)∈A with respect to * is `(1/a,−b/a)`∈A−1.
(iii)Now let us find the inverse of elements `(5, 3) and (1/2,4)`
Hence , inverse of `(5,3) is (1/5,3/5)`
And inverse of `(1/2,4) is (2,-4/(1/2))=(2,-8)`
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