I want to know whether following are true or false:
for any given natural number $n$, $T>0$ a rational, suppose that $Q_1, \cdots, Q_n$ are $m\times m$ matrices with rational entries, $t_1, \cdots, t_n$ are positive real numbers with $t_1+\cdots+t_n=T$,
the number $\alpha \cdot \Pi_{i=1}^n e^{Q_it_i} \cdot \beta$ is
(1) NOT a rational number; (2) is NOT a rational number almost surely? Here, $\alpha$ is arow vector and $\beta$ is a column vector, both with rational entries.
I know that for n=1, (1) (and hence (2)) is true, which can be proved by Lindemann-Weierstrass theorem (note that T is a rational). However, I do not know if $n=2$.
Thanks!