Prove a closed linear group is not unimodular

Question

I am trying to solve Exercise 9.4.2 in the book "Ergodic Theory with a view toward Number Theory" by Einsiedler and Ward. The problem is to show that the closed linear group $$T=\left\{\left( \begin{array}{cc} e^{t/2} & s\\ & e^{-t/2} \end{array} \right)\mid s,t\in\mathbb R\right\}$$ does not contain a lattice (a discrete subgroup with a fundamental domain of finite left Haar measure). My idea is to use Proposition 9.20 in the book, which says (among other things) that a closed linear group that contains a lattice must be unimodular. Thus, I simply need to show that $T$ is not unimodular. However, I feel as though this requires me to know something about the left Haar measure on $T$. Is there any easy way to compute this Haar measure? Alternatively, is there a nice way to show that this group is not unimodular? Thanks in advance.