(1) LetG = {(ad)and recall matrix multiplication(ªd) · (& & ) = (²be=1},: a, b, c, d E R, and ad-bc=ab' + bd'aa'+beca' + de cb' + dd'Show that (G,) is a group. (This group is called the speciallinear group of degree 2, and usually denoted as (SL(2, R),.).)

Let We have to show that is a group. Here '.' denotes the usual matrix multiplication of…

Answered: (1) Let G = {(ªd) : a,b,c,d € R, and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

(1) Let
G = {(ad)
and recall matrix multiplication
(ªd) · (& & ) = (²
be=1},
: a, b, c, d E R, and ad-bc=
ab' + bd'
aa'+be
ca' + de cb' + dd'
Show that (G,) is a group. (This group is called the special
linear group of degree 2, and usually denoted as (SL(2, R),.).)

Expert Solution

Step 1: ''Introduction to the solution''

Let $G equals open curly brackets open parentheses table row a b row c d end table close parentheses colon a comma b comma c comma d element of straight real numbers comma space a n d space space a d minus b c equals 1 close curly brackets$

We have to show that $open parentheses G comma space. close parentheses$ is a group. Here '.' denotes the usual matrix multiplication of matrix.

(1) G is closed under '.' :

Let $open parentheses table row a b row c d end table close parentheses comma open parentheses table row cell a apostrophe end cell cell b apostrophe end cell row cell c apostrophe end cell cell d apostrophe end cell end table close parentheses element of G$ be two arbitrary element.

Then, $open parentheses a d minus b c close parentheses equals 1.......... left parenthesis 1 right parenthesis space space$

and $open parentheses a apostrophe d apostrophe minus b c apostrophe close parentheses equals 1.......... left parenthesis 2 right parenthesis$

Now, $open parentheses table row a b row c d end table close parentheses. open parentheses table row cell a apostrophe end cell cell b apostrophe end cell row cell c apostrophe end cell cell d apostrophe end cell end table close parentheses equals open parentheses table row cell a a apostrophe plus b c apostrophe end cell cell a b apostrophe plus b d apostrophe end cell row cell c a apostrophe plus d c apostrophe end cell cell c b apostrophe plus d d apostrophe end cell end table close parentheses$

$N o w comma space open parentheses a a apostrophe plus b c apostrophe close parentheses open parentheses c b apostrophe plus d d apostrophe close parentheses minus open parentheses a b apostrophe plus b d apostrophe close parentheses open parentheses c a apostrophe plus d c apostrophe close parentheses space space space space space space equals a d left parenthesis a apostrophe b apostrophe minus b apostrophe c apostrophe right parenthesis minus b c left parenthesis a apostrophe b apostrophe minus c apostrophe d apostrophe right parenthesis space space space space space space equals left parenthesis a d minus b c right parenthesis left parenthesis a apostrophe b apostrophe minus c apostrophe d apostrophe right parenthesis space space space space space space equals 1.1 equals 1 space space space left square bracket b y space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis right square bracket$

This shows that G is closed under '.'