Thread: Matrix Question

Consider the 2x2 matrices

A= [(rt3)/2 1/2 ]
[1/2 -(rt3)/2]

B= [-1/2 (rt3)/2]
[(rt3)/2 1/2 ]
in the multiplicative group of non-singular 2x2 matrices with entries in R. Let G be the group generated by A and B which you may assume is order 8.

i) List the non-identity elements (other than A and B) of G, both as explicit 2x2 matrices and as products involving A and B

Now I am getting a bit confused as to what these elements are. Are they AB, BA, A^2, B^2. Is this all of them as I am not sure that since the order of G is 8 there should be 8 elements and with the four above plus the three I do not need (identity, A,B) I still need 1 more.

Also what do they mean by express as both explicit 2x2 matrices and as products involving A and B?

For example:

AB=[(rt3)/2 1/2 ][-1/2 (rt3)/2]
[1/2 -(rt3)/2 ][(rt3)/2 1/2 ]

=[0 1]
[-1 0]

Have I expressed this as both an explicit 2x2 matrix, as well as a product involving A and B.

Hope you can clear this question up for me
(Sorry if the matrices are a bit dodgy, but it wouldn't let me write it in a straight line)

Many thanks

Bex

Originally Posted by bex23

Consider the 2x2 matrices

,

,
in the multiplicative group of non-singular 2x2 matrices with entries in R. Let G be the group generated by A and B which you may assume is order 8.

i) List the non-identity elements (other than A and B) of G, both as explicit 2x2 matrices and as products involving A and B

Now I am getting a bit confused as to what these elements are. Are they AB, BA, A^2, B^2. No. For a start, A^2 and B^2 are the same (they are both equal to the identity matrix).

Is this all of them as I am not sure that since the order of G is 8 there should be 8 elements and with the four above plus the three I do not need (identity, A,B) I still need 1 more. There are indeed several more.

Also what do they mean by express as both explicit 2x2 matrices and as products involving A and B?

For example:



Have I expressed this as both an explicit 2x2 matrix, as well as a product involving A and B. Yes, that's exactly the sort of thing that is meant. Notice that a group has to have an identity element (which in this case has to be the identity matrix). The product AB is not the identity, so try taking powers of it. You'll find that (AB)^2 is –I and (AB)^4=I. That should give you some idea of the sort of matrices that constitute this group.

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