Question

Please solve #22 with step by step.

I need to understand clearly from your solution. also, please use the Hint.

When you provide the solution, please write it clearly to be recognizable.

22. Find Z[GL2(R)]. 6. (Center of the matrix group.) Solve Problem 22, Section 2.3, asking you to find the center of the group GL2(R). Hint: Let A= ( 2) be in the center, and write down explicitly the condition that AB = BA for all matrices B= (%) € GLĄ(R)

Answer 1

Ans the group Glalies defined as GLA (IR) = A ( a b] : a, b,c,d flR? Sitial Accorting to definition of center of any group a. nofte ofhe center of group a is denoted and defind of z (a) = hachi ab=barben? = callection of those elemente of Group a which commute every element of group G. Hem G = Ghz (IR) = Sut of all invertible matriay, of orter 2x2 then ill have to find such type of matricy whics which comupha elry matrics. AE 2 (a) is AB = B A E B G G =GIR) ! lie

In general matrix multiplication is not Commutahim lip H A,B EG4 (IR) 1 AB A BA in general How let A, BE GLA (IR) on A=1 a 67 B = 1 11 20 n 3 then AR. axtbz I cxtd2 1. ay tbway (4tdW. bxtdy BA= (axtly HE bzfdw | a2 + We y AEZ (GLER )) { center of aroup}. then AB = BA VBA GLA (IR) ve AB. BA=0 [bzocy .ay tow box of dy 7 ro *** 76037 I lo cx +J2 - azt we C4-62 cyabz bz .cyzo 2 b=c=0 4 289 is nonzero Cet d2 -az thc - 0 (d-a92 zo sinu brezo from the abou Here also if ato thang-al=0 C = d=a

z WJ - - i B=1 ] is non ziro malmix with non zero entries » A become as c z (hL2R)) = 689 &J : aelRS lie center of GLIR) is scallection of all schler matricu.