Question

The set of invertible 2 by 2 matrices is denoted GL(2, R) = { (a b; c d): a,b,c,d all element of R and ad - bc cannot = 0}.

a) Show that the matrix multipication is a binary operation on GL(2,R). In other words, if A,B are elements of GL(2,R), show that AB is an element of GL(2,R).( Hint: det(AB)=det(A)*det(B)

Answer 1

So you need to show that det(AB)=/=0

Answer 2

no we are showing if it is a binary operation i dont know what that means

Answer 3

I didn't either, but luckily it said right after: "In other words, if A,B are elements of GL(2,R), show that AB is an element of GL(2,R)"

Answer 4

hmm yes so could we just set A=(a1 b1; c1 d1) and B=(a2 b2; c2 d2) and then multiply them togehter?

Answer 5

No, no need for that. We need to show something for det(AB), from the hint we know det(AB)=det(A)*det(B).

Answer 6

What do we know about det(A) and det(B)?

Answer 7

that a1d1 - b1c1 cannot = 0 as well as a2d2 - b2 c2

Answer 8

Right, so det(A) isn't zero and det(B) isn't zero. So what do we know about det(AB)?

Answer 9

that obviously cannot equal zero

Answer 10

but dont we have to show it some how i tried multiply them or what ever and got a1a2d1d2 +b1b2c1c2 - a1b2c2d1 - a2b1c1d2 and then i got stuck

Answer 11

No we don't. See that's the problem you're having, you're trying to go those elements, but that's only making it harder for no reason.

Answer 12

We know det(AB) isn't zero, therefore AB is in GL(2,R)

Answer 13

GOT IT :D

Answer 14

cool

Answer 15

from that formula i got a2d2(a1d1 - b1c1) + b2c2(b1c1 - a1d1)

Answer 16

cannot equal zero and then finally got a2d2 - b2c2 cannot equal zero

Answer 17

could you also help me out with this second part as well

Answer 18

That's one way to do it, but you're making it a lot harder than necessary.

Answer 19

so could we say since detA cannot equal zero and det B cannot equal zero that must imply det(AB) cannot equal zero?

Answer 20

That's right.

Answer 21

hmm alrightt could you also help me out with this second part to this question
it says

Answer 22

sure

Answer 23

Show that the matrix multipication is associative. In other words if A,B,C,D are element of GL(2,R) show that A(BC)=(AB)C

Answer 24

though trivial..you should also point out that since A is 2x2 and B is 2x2 then AB is 2x2

Answer 25

could we just introduce those matrices and multiply them and see if the right side is equal to the left side or is there more that we have to do

Answer 26

@Zarkon thanks a lot i would try remembering to write that just perparing for this midterm

Answer 27

You could compare both sides, but I think this is easier. Matrix multiplication is associative in the set of all matrices, so it is also associative in GL(2,R) (a subset of the set of all matrices.)

Answer 28

alright and to find the identity for this matrix, is the I2 matrix valid?

Answer 29

You mean the identity in GL(2,R)?

Answer 30

yeah

Answer 31

like would that be valid I2 = ( 1 0; 0 1)

Answer 32

That is the identity. The same goes here, because I2 is the identity in the set of all 2x2 matrices, it is also the identity in the subset GL(2,R)

Answer 33

alright perfect thanks a lot man really appreciate it :)

Answer 34

You're welcome.