If $P e^{x} = Q e^{-x}$ for all real values of $x$, which one of the following statements is true?

Given that the identity holds for all real values of $x$:

1. Substitute $x = 0$:

   $$P e^0 = Q e^0 \implies P = Q$$

2. Now, substitute $Q = P$ back into the original equation:

   $$P e^{x} = P e^{-x} \implies P(e^x - e^{-x}) = 0$$

3. Since this must hold for all $x \in \mathbb{R}$, choose any non-zero value, for instance $x = 1$:

   $$P(e^1 - e^{-1}) = 0$$

Since $e - e^{-1} \approx 2.718 - 0.368 \neq 0$, it follows that:

Since $P = Q$, we also have $Q = 0$.

Therefore, the only statement that is true is $P = Q = 0$ (Option A).