Stuck on an example of why a Matrix lie algebra needs to be closed with respect to GL(n,C)

Question

Can someone please help me on this example.

$$ G = \left\{ \begin{pmatrix} { e }^{ it } & 0 \\ 0 & { e }^{ 2\pi it } \end{pmatrix},\quad t\in\mathbb{R} \right\} $$

G is a matrix group, but not a matrix Lie group, why not?

I cannot think of a sequence which is in G and converges in GL(n,C) but not in G.

Thank you in advance.

Harch

Answer 1

Hint $-I\notin G$ can you see that????????????????