Given elts of the group SL(2,R), find |A|, |B|, and |AB|.

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mcwang719

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Consider the elements:

A= [first row 0 -1, second row 1 0] (matrix)

B= [ first row 0 1 ,second row -1 -1]

...from the group SL(2,R). Find |A|, |B|, and |AB|.

I know that |A| is the order of A. I read the section and can't find anything on order for matrix. How would I find the order of a matrix? Does it have something to do with the determinent?

Thank you!
 
SL(2,R) is a subset of the group GL(2,R). i think it's the way they denote a matrix.
 
Well O.K. now what is GL(2,R)?
I know it is hard to believe but notation is not ever standard.
 
I found this in the book GL(2,R)={|abcd| (2x2 matrix) a,b,c,d E R, ad-bc does not equal 0} The E represents the element of.
 
Now what does the text say SL(2,R) is?
 
the book doesn't give a definition but from reading SL(2,R) is a group with the operation of Matrix multiplication, Identity of [first row 1 0 second 0 1] (matrix), form of element is [first row a b secon c d]. Inverse is [ first row d -b second -c a]. it's not abelian and ab-bc=1. i have the solutions which is |A|=4, |B|=3, and |AB|=infinite.
Thanks.
 
What is the textbook? I may have have it.
 
SL(n,R) is the Special Linear group of nxn matrixies with a determinant of 1. GL(n,R) is the General Linear group of nxn matrixies with non-zero determinants.

The order of ANY element g in a group is defined as the smallest positive integer n such that g^n=e, where e is the identity of g. If no such integer exists, g is said to have infinite order.

In your case the idenity is I<sub>2</sub>, the 2x2 identity matrix. What power can you raise A to that will yield the idenity matrix? What about B? Now try the product AB. The idea of this exercise (said by my professor last semester) seems to be that if two distinct elements have a finite order, their product still may have an infinite order (which you should verify).
 
i got it!!!! i reread you're last paragraph and finally got it A^4= identity. yay!!!! thanks everyone.
 
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