The discussion revolves around demonstrating that the general linear group GL(3,R) with matrix multiplication is not an abelian group. Participants are tasked with showing that the group does not satisfy the property of commutativity.
Multiple participants have attempted to provide examples of matrices that do not commute, presenting calculations to support their claims. While some participants express satisfaction with the ease of finding such examples, there is no explicit consensus on the finality of the discussion.
Participants are operating under the constraints of homework guidelines, which may limit the depth of exploration or the types of examples considered. There is an emphasis on finding non-commuting matrices without specific restrictions on their forms beyond avoiding both being diagonal.
mikael27 said:Homework Statement
Show that the general linear group GL(3,R) with matrix multiplication is not an abelian
group.
Homework Equations
The Attempt at a Solution
A group to be abelian we have to show that it satisfies the Commutativity.
a*b=b*a
How are we going to show that the general linear group GL(3,R) does not satisfies the Commutativity?
mikael27 said:Take
A=
0 -1 0
1 0 0
0 0 1
B=
1 0 0
0 0 -1
0 1 0
A*B=
0 0 1
1 0 0
0 1 0
B*A=
0 -1 0
0 0 -1
1 0 0
So the two matrices do not commute as A*B=B*A and hence it is not abelian