Let S denote the set of positive real numbers. Define operations and by the equations and x + y := xy xoy :=xlny (a) Prove that S with the operations and is a ring. What are 0s and 1s? (b) Is S commutative? (c) Find the units and the zero divisors in S. For every unit u € S, find its inverse v, i.e., v € S such that u Ov=vOu=1s.

Determine the value(s) of a such that X = [-6 [ 2 1] -3 6 Note: If there is more than one value write them separated by commas. -3 6 -1 -2 -1 x = 0

Compute the adjugate of A and use it to find the inverse, A-¹. 4 -1 07 A = 0 12 -3 1

13 4 3 1 Let T:R2 - R2 be multiplication by A = Determine whether T has an inverse; if so, find If you choose "has not" as the first answer, don't answer the second question. If you choose "has" as the first answer, express your answer to the second question in horizontal notation. an inverse; T(x1, X2) = (X1 – X1 + X2 ) Click if you would like to Show Work for this question: Open Show Work