11. Let A=QxQ.Let * be a binary operation on A defined by : (a,b)*(c,d) = (ac,ad+b).Find i) Identity element of (A,*) ii) the invertible element of (A,*).

As A is defined on rational numbers
A = QxQ
So (a,b)*(c,d) = (ac,ad+b)

1) for identity element, we have a*e = a
Let a = (a1 ,a2 ) and e =(e1 ,e2)
So a*e = ( a1e1 , a1e2 + a2)
This should be equal to a = (a1 ,a2)
So e = (1,0 ) satisfies this condition.
As ( a1e1 , a1e2 + a2) = ( a1(1) , a1(0) +a2 ) = (a1 ,a2)

2) For invertible element, we have a*b = e = b*a
As we have found e already,
So as a = (a,b) , and b = (x1,x2)
So a*b = ( ax1 , ax2 + b) = (1,0)
This will satisfy when x1 = 1/a , and x2 = -b/a
So the invertible element = (1/a ,-b/a)
 

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