(a,b)∗(c,d)=(ac,ad+b)(1) identity element
let (e,f) be the identity element .
Then (a,b)∗(e,f)=(ae,af+b)=(a,b)
[a∗e=a, where e is the identity element ]
(ae,af+b)=(a,b)
⟹ae=a and af+b=b
⟹e=1 and f=0
(1,0) is the identity element .
(2) invertible element
Let (c,d) be the inverse of (a,b)
then
(a,b)∗(c,d)=(1,0)[a∗b, e is the identity ]
⟹(ac,ad+b)=(1,0)
ac=1;ad+b=0
c=1a,d=−ba
So, (1a,−ba) is the inverse.
The inverse of (5,3) is (15,−35)
The inverse of (14,4) is (4,−16)