Question

please provide with full working solution. thank you

Consider the set B of all 2 x 2 matrices of the form {C 9 b a B a, b e R -b a and let + and . represent the usual matrix addition and multiplication. (a) Determine whether the system B = (B, +,.) is a commutative ring. (b) Determine whether the system B = (B, +, .) is a field. T

Answer 1

a,be Given: B2 abelian yp. A, - Closurs Propcty: Let A, ER hen ArtA b'b Bis clased under additian o atices Associadtive pro perty: We knw that addi fign omatatces tollawS soiative law A., AAe B At (At A) A+A)+ A foY all Existance o addihive idanhty: (pating azo, o) tLere exisls GB Such hat for auy matn Aa -S A+oaL A identty *.O is the ackdihive Existance a addtve inverse here exists amathx for eveny matix A= Fuch fha AtA) (-A)+ A = 0 -A is the adlduhve inverse A ER We knnu hat addihm f wmahicey s Canutafie Conmuthive Progety For A A, A,tAz l

-b2 for all A, AER At A AtA abelian grp. (B+) is an R (Bis a enir: product awy huro 2x 2 mathx Clsure properhy: We know that the matices is aso R$S G-4 9,6+ A, = Nov, Let A * A A 2 = b'b b-71 Bis clazed nder mlhpiatirn watices Associatin Propert natices we know at mltpliehin Adow assorite property ACAAA, A) A fur al A, A, (B,) Semiap we know tat for ay tree mat hcep A,A2 A orduy 1x A,(AtA) A, Az+ A, A2Let dismbti laus (A2t A2) A, A Art AzA, frisld aishi butive law aud .. FhesR laws are true for al matrices ,A, A ao- (2t,) is a A, A a, b Nou, -h 4 - b a29- -26-9h 4-

AA A, AEA all A, A AA for is a commutative ring is a commtetve ring (Pudtng there ekists I = Such thar AI for all A G A with identity B s rina Let A =ab Atc Ato at deast nne he ues a and ts non zero det CA)= | a =atb o b1-1 .. A is inverhs le and A- adi A det CA) at a a E B ath nch hat AA AA -I (B,t,) is a ficld