Examples of $e^A e^B \neq e^C $

Question

Can anyone give me an example where for matrices $A$ and $B$, there is no matrix $C$ such that

$$ e^A e^B = e^C ? $$

Answer 1

HINT:

$$e^A= \left( \begin{matrix} -1 & 0 \\ 0 & -1 \end{matrix} \right ) \\ e^B= \left( \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix} \right )$$

$e^A \cdot e^B = \left( \begin{matrix} -1 & 1 \\ 0 & -1 \end{matrix} \right )$ is not the exponential of a real matrix.

(Note that any invertible matrix is the exponential of a complex matrix)