Show that $e^{At}B = B e^{At}$ if $AB= BA$ Let $A$ and $B$ be two matrices. How can I show that $e^{At}B = B e^{At}$ if $AB= BA$ and then conclude that $(d/dt) e^{At} e^{Bt} = (A+B) e^{At} e^{Bt}$? I have no idea how to do this. 

More generally, if $AB = BA$ then $f(A)g(B) = g(B) f(A)$ for any polynomials $f$ and $g$, and then by continuity for any entire functions (you can do somewhat better than that, but that's enough for this case).
