Question

1,(Z) = { a bla, b, c, d. Let M2(Z) = a, b, c, d e Z} with matrix addition and multiplication. Which of the following is true: it is a commutative ring with unity it is a ring with unity but not commutative it is a ring without unity and not commutative it is a commutative ring without unity Question 4 Which of the following statement is true about the ring of integers (with usual addition and multiplication) the integers are a field the integers are an integral domain the integers are both a field and an integral domain

Answer 1

Ques) Let M₂ (Z) = { [a b] la, b, c, d E Z} Aus (6) Ring with unity but nou - commutative. Let Mr = {m: M is 2x2 matols} Properties Under Addition:- 0 Clasure Laui AS sum of two 2x2 matrices is again a matrix of type 2x2. & M₂ is closed wrot mattis addition. & Associative Lain At B+C) = (A+B) + ( for A,B,GEM, This addition of matrices in an associative composition. ③ Existence of Inverse – Gor each matrix AtMy, there exist a matrix – A EM₂ such that - A+A = Ohxh = A + (-A) Thus, the inverse exists. Existence of identity. Null matrix o of type nan belongs to My and Onxn + A = A + Onxn=A V A E Ma Hence, identity Mataix exists. 6 Commutative lawin for allt A, BE M23 we have A+B=B+A :. Addotion of matrices is conanntative Properties Under Multiplication: @ Closure how for multiplication - The product of two Rx2 matrices is a matrix of type 2x20 : A.BGM a for all A, BE M2. @ Associative Laut Matrix multiplication is associative A CB CS - AB e for all A, B, C EM..

♡ Existence of identity' - Unit matrix I of type 2x2 an dement of M2 - As I.A - AIAK A EM2 Identity matrix of multiplication exists. ② Distributive Lane! – A, B, C EM2 , we have A B+C) = A.B TA.C (B+C). A = B. A&C.A. Hence Mn is a ring with unity in is a ring with writy Wr.t the additim & Multiplication. As the matrix multiplication is not in general commutative, therefore the ring is a non commutative So M₂ (2) is a ring with unity but non-commutative, gues y its field is a connectative division ring. set of Integers is not a field as lit lacks multiplicative inverses without multiplicative Inverden division may not be impossible. The anly emersible dements of this rings are 2 and -1. li Integrale domain is a comunitative sing without zero - divisors. Means it to both ab=0 2 azo orbco Let a, b G Z , with a b to If a bao 2 either azo or beo 2) set of Integers are Tute gral domain 2 CZ, t, e) is a Ibat egral domain. 26 The Integends are an Integral Domain