General Linear Group not Abelian

[mikael27]

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?

 

[Dick]

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?

Find two matrices that don't commute. Just try a few. Just don't make them both diagonal.

 

[mikael27]

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

 

[mikael27]

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

 

[Dick]

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

That's just fine. Wasn't that easy?

 

[mikael27]

Yes easier than i expected...Thanks !