Question

Let $A=Q\times Q$ and let* be a binary operation on A defined by $(a,b)\ast (c,d)=(ac,b+ad)for(a,b),(c,d)ϵA.$ Determine, whether * is commutative and associative. Then, with respect to * on A
Find the invertible elements of A.

Answer 1

(i) For any (a, b), (c, d) and (e, f)$\in$A, we have ${}^{\prime }{\star }^{\prime }$.

$\left\{\right(a,b)\ast (c,d\left)\right\}\ast (e,f)$

$=(ac,b+ad)\ast (e,f)$

$=(ace,b+ad+acf)$

and

$(a,b)\ast \{cc,d)\ast (e,f)\}$

$=(a,b)\ast (ce,d+cf)$

$=(ace,b+ad+acf)$

So, $\left\{\right(a,b)\ast (c,d\left)\right\}\ast (e,f)=(a,b)\ast \left\{\right(c,d)\ast (e,f\left)\right\}$ for all a, b, c, d $\in Q\times Q=A$

$\Rightarrow "\ast "$ is associative on A.

(ii) Let (a, b) be invertible element on A

$(c,d)\in A$ such that

$(a,b)\ast (c,d)=(1,0)=(c,d)\ast (a,b)$

$\Rightarrow (ac,b+ad)=(1,0)$ and $(ca,d+bc)=(1,0)$

$\Rightarrow ac=1,b+ad=0$ and $ca=1,d+bc=0$

$\Rightarrow c=\frac{1}{a},d=-\frac{b}{a}$ if $a\ne 0$

Thus $(a,b)$ is invertible element of A.