Question

Question 2. Recall that a monoid is a set M together with a binary op- eration (r,y) →エ. y from M × M to M, and a unit element e E/, such that: . the operation is associative: for all x, y, z E M we have (z-y): z = the unit element satisfies the left identity axiom: for all r E M we have the unit element satisfies the right identity axiom: for all a EM we Let K be a field, and let M2(K) denote the set of 2 × 2 matrices with (y z): e-x=x; and have x-e=x. coefficients in K 2.1 Using the operations of K, give the definition of the "matrix multipli- cation" map (A, B) → A-B frorn M2(K) × M2(K) to M2(K). (Hint: you already know how to multiply matrices, just write it down formally in terms of the operations.) (you need to say what the unit element is). commutative monoid? 2.2 Prove that M2(K) with the operation of matrix multiplication is a monoid 2.3 What is the definition of "commutative monoid"? Is this monoid a uestion 3. Suppose (M, e) is a monoid. Let M C M denote the subset onsisting of elements a E M such that there exists y e M with y-e (y is called a "left inverse"). Let MR C M denote the subset consisting of elements x E M such that there exists z E M with x , z = e (z is called a right inverse") 3.3 S invertible. uppose M is a fipite set. Show that if a has a left inverse then it is Question 4. App Questio 3 to the monoid M2(K) o Question scribe the group of inverore elements. That is called GL De

Answer 1

Let A = [a c b d] a, b, c, d elementof B = [p r q s] p, q, r, s elementof k M_2 (k) times M_2 (k) rightarrow M_2 (k) (A, B) rightarrow A middot B = [ap + br cp + dr aq + bs cq + ds] Since K is a field, therefore A middot B elementof M_2 (K) Associative Let A = [a_1 c_1 b_1 d_1] , B = [a_2 c_2 b_2 d_2] and C = [a_3 c_3 b_3 d_3] are in M_2 (k) AB = [a_1 a_2 + b_1 c_2 c_1 a_2 + d_1 c_2 a_1 b_2 + b_1 d_2 c_1 b_2 + d_1 d_2] \r\n (AB) C = [(a_1 a_2 + b_1 c_2) a_3 + (a_1 b_2 + b_1 d_2) c_3 (c_1 a_2 + d_1 a_2) a_3 + (c_1 b_2 + d_1 d_2) c_3 (a_1 a_2 + b_1 c_2) b_3 + (a_1 b_2 + b_1 d_2) d_3 (c_1 a_2 + d_1 c_2) b_3