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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 - CBSE (Commerce) Class 12 - Mathematics

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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,  yQ such that X*E=X=E*X, XA

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, nQ

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/and b+an=0

n=b/a   

Also,

Considering (ma, n+mb)=(1, 0)

ma=1

m=1/and n+mb=0

n=b/a          (m=1/a)

The inverse of (a, b)A with respect to * is `(1/a,−b/a)`A1.

(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)`

  Is there an error in this question or solution?
Solution 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 Concept: Concept of Binary Operations.
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