So here is my question,
I would like to prove,
If $R,S\in \mathcal M_{n\times n}(\mathbb R)$ are matrices such that, $$e^{t(R+S)}=e^{tR}e^{tS},\;\forall t\in\mathbb R$$ Then, $$RS=SR$$
And here is what i did,
$$e^{t(R+S)}=e^{tR}e^{tS}$$ $$\Leftrightarrow0=e^{t(R+S)}-e^{tR}e^{tS}$$ This implies,
$$0=\frac{d}{dt}(e^{t(R+S)}-e^{tR}e^{tS})$$ $$\Leftrightarrow 0=(R+S)e^{t(R+S)}-Re^{tR}e^{tS}-e^{tR}Se^{tS}$$ $$\Leftrightarrow 0=Re^{tR}e^{tS}+Se^{tR}e^{tS}-Re^{tR}e^{tS}-e^{tR}Se^{tS}$$ $$\Leftrightarrow 0=Se^{tR}e^{tS}-e^{tR}Se^{tS}$$ $$\Leftrightarrow Se^{tR}e^{tS}=e^{tR}Se^{tS}$$ $$\Leftrightarrow Se^{tR}=e^{tR}S$$ By differentiating on both sides we obtain,
$$SRe^{tR}=Re^{tR}S$$ Setting $t=0$ proves finally the claim,
$$SR=RS$$
Could someone look over my prove an tell me if it is correct, or if not give me a hint where I made a mistake? Thanks!