Let $A$ and $B$ be two same dimensional matrices and consider one parameter groups $e^{aA}$ and $e^{bB}$ (under product) where $a$ and $b$ are arbitrary reals.
I want structure of smallest group which contains this two groups (under product).
Is it true that answer is the set of all matrices of form $e^{aA+bB}$ where $a$ and $b$ are arbitrary reals?
If it is true even under some circumstances have you any proof?