A counterexample is given by 
$$A=\begin{bmatrix}
 -1 & a \\
 a & -1 \\
\end{bmatrix}$$
with (say) $a=2$. 

Indeed, then the sum of your first series is 
$$c(t):=\frac12\begin{bmatrix}
 c_+(t) & c_-t) \\
 c_-(t) & c_+(t) \\
\end{bmatrix}$$
with 
$$c_\pm(t):=\cosh t\pm\cos \left(\sqrt{3} t\right)\ge1-1=0$$
for real $t$, and the sum of your second series is 
$$s(t):=\frac12\begin{bmatrix}
 s_+(t) & s_-t) \\
 s_-(t) & s_+(t) \\
\end{bmatrix}$$
with 
$$s_\pm(t):=\sinh t\pm\frac{\sin \left(\sqrt{3} t\right)}{\sqrt{3}}\ge0$$
for real $t\ge0$, because $s_\pm(0)=0$ and $s'_\pm=c_\pm\ge0$.