Does the group of 3×3 matrices with exponential diagonal have a center?

[arz2000]

Hi all,
can you show that the group of all 3 by 3 matrices
[e^t 0 u
0 e^xt v
0 0 1]
where t, u, v are in C (complex numbers) and x is in R (real number)
has no center?

Regards

 

[arz2000]

I mean all 3 by 3 matrices with the following rows
(e^t, 0, u)
(0, e^(tx), v)
(0, 0, 1).

 

[matt grime]

You can't, since it does have a centre - every group has a centre, possibly trivial (as it is in this case).

You just write down two matrices, suppose the commute and show that this implies that they are both the identity matrix.