Question

ABSTRACT ALGEBRA: pROVE THAT THE SET OF ALL 2X2 MATRICES WITH ENTRIES FROM r AND DETERMINANT +1 IS A GROUP UNDER MATRIX MULTIPLICATION. @ash2326 @robtobey @whpalmer4 @wio @Compassionate @shamil98 @beccaboo333 @nincompoop @wolfe8 @Zale101 @Ashleyisakitty @e.cociuba @Isaiah.Feynman @kewlgeek555 @tester97 @Gabylovesyou

Answer 1

Sorry I'm not good with math.

Answer 2

Sorry im barely passing alg 2 i doubt i could help you with that.

Answer 3

First, let's let G be all of the 2x2 matrices

Answer 4

Now, do you know the definition of a group under matrix multiplication?

Answer 5

Yep! It has to have closure, associativity, an identity element, and an inverse! (Took linear algebra last semester so I know all these cool terms lol)

Answer 6

So our goal is then to prove all of them.

Answer 7

Let's start with the easy one.

Answer 8

Can you please tell me what the notation GL(2,R) is in abstract algebra?

Answer 9

That means the set of all invertible matrices in Real number

Answer 10

Correction: the set of 2x2 invertible matrices over real number

Answer 11

Now, one of the definitions was that it has to include the identity matrix, right?

Answer 12

@science0229 Now what does SL(2,R) mean then?

Answer 13

SL(2,R) is the set of 2x2 matrices with determinant of 1 over real number

Answer 14

How do you know that? Haha

Answer 15

I know many random things.
I was studying matrices. I got curious. I went in a little bit deep, and that came up...

Answer 16

@science0229 How is {1,2,3,4} under multiplication modulo 5 a group?

Answer 17

Hey. I don't know that deep!
I'm just an 8th grader with curiosity.

Answer 18

Here how you do it
1) closure
det (A B)= det(A) det(B) =1x1 =1
2) If A is in G, then the inverse of A is
\[
det(A^{-1}) = \frac 1 {det(A)}= \frac 1 1=1
\]

Answer 19

The identity matrix I belong to G since det(I) =1
That is all what you need to show. Since Multiplication of matrices is associative