2 Let be the following binary operation on on R2 a b c d ac

2. Let be the following binary operation on on R2: (a, b) c, d) ac, ad b) (a) Does (R2, have an identity element? (b) Find the inverse of (a, b) in (R2,

Solution

(a) Yes, it has the identity element

Let the identity element be (x1,x2), then it must satisfy the equation

ax1 = a

ax2 + b = b

Hence we get x1=1 and x2=0

Therefore the identity element is (1,0)

b) The inverse of (a,b) will map it to (1,1). let the inverse of (a,b) be (a1,b1)

aa1 = 1

ab1 + b = 1

a1 = 1/a

b1 = (1-b)/a

Hence the inverse elemet will be (1/a, (1-b)/a)