Question

Show that the set of matrices of the form where a, b ∈ Q is a field under the operations of matrix addition and multiplication. (abstract algebra)

a 6 26 a

please show the following axioms (closure, identity, associative, distributive, inverse, and commutative) for addition and multiplication

Answer 1

det, 5= { (a 6); a, 668) det, A= (ar 6.) ES and B= (26 a-las bres (26, a E an) where, as, be, as, b, Ed i Now, A+B = /auta bitbe | ES & ABES (2/6, +b) a tag) where (a1 +92), (6 + be) e 8 So, s is closed under addition. i) Let, c= ( 2 bojes where, az, by eg Clearly, A+(B+C) = ( A+B) + e + A,B,C ES So, s holds ansocialive property under addition. ii) Clearly. (%)ES and on abi ) + ( )=(80)+(20%, )= ( 4 ) So, A+ 0 = 0+ A = A VAES wher, 0-(3) So, l . ) is the identity element of s under addition, ivy As, a 6.1.1-01 -bi -1-04 -bila bilo " As loh alt 1-261 a)-(-26, -at (2b, aloo a where, -a,ob, E8 Therefore, for each element of S. has inverse in S. a belas bel- / bala bil v) As, (a ) + (26. a) = (abg an) + (26, ail So, A+B=B+A & ABES So, sin commuta live under addition Therefore, (5,t) is a obelian group.

wi) A.B = 1 9 bila bol (2b, a (ab al Ia a + 2b, be abetab, - (210462 + a₂b,) qaz +26, bel TES where for a₂ + 2 b, ba), (arba +Qz6.) Eg is is cloned under multiplication. will clearly, Albe) = (AB). & A, B, CES is is ansociative under matrix multiplication, vül Clearly, A.(B+C) = A.B + A.C f (B+C). A = B.A+CA V A,B,CES Therefore, (5,+,) forma a suing. ix) Ket, I - ) and A. J = 1.A : ) 126, a la aux tably = a f any + bir = bi Solving we get, x= 1egl y =of 9 So, loi) is the ideality element of s under matrix multiblication . x det A lan bila an 267 #0 & a, b, eg So, À exints Now, Il-biles Ä = -267 (-26, an). Therefore, each von zero element of Sis a unit. xi) Now, AB = ca a az +2 biba ar bgt azbil (210462_+Qzbe) au Q2 +2.b, bax! So, (5,.) in abelian grouß undan meliblication, Hence, (5, +, .) is a field. 1 = BA & ABES