Inverse of an element

Consider the binary operation * and o defined by the following tables on set S={a,b,c,d}.( FIGURE) Show that both the binnary operations are commutative and associative.Write down the identities and list the inverse of elements.

Let A=QxxQ , where Q is the set of all rational numbers and '**' be the operation on A defined by : (a,b)**(c,d)=(ac,b+ad)" for "(a,b),(c,d)inA . Then, find : (i) The identity element of '**' in A (ii) Invertible elements of A and hence write the inverse of elements (5, 3) and ((1)/(2),4) .

Inverse of a matrix

Let * be an associative binary operation on a set S with the identity element e in S. Then. the inverse of an invertible element is unique.

Transpose and Inverse of a Matrix

Let ^(*) be a binary operation on Q-{-1} defined by a*b=a+b+ab for all a,b in Q-{-1}. Then,Show that every element of Q-{-1} is invertible.Also,find the inverse of an arbitrary element.

If A is a 2 xx 2 scalar matrix and 7 is the one of the elements in its principal diagonal, then the inverse of A is_____.