Jtdylktuy

I have a reflection matrix $A=begin{pmatrix}-frac{1}{2} & frac{sqrt{3}}{2} \ frac{sqrt{3}}{2} & frac{1}{2} end{pmatrix}$ and a linear, autonomous, and homogeneous system $frac{doverrightarrow{w}}{dt} = Aoverrightarrow{w}$, $overrightarrow{w}(t) = begin{pmatrix} x(t) \ y(t) end{pmatrix}$.

I've noticed that $left{begin{pmatrix}frac{1}{2} \ frac{sqrt{3}}{2}end{pmatrix},begin{pmatrix} -frac{sqrt{3}}{2} \ frac{1}{2}end{pmatrix}right}$ is an eigenbasis with eigenvalues $lambda = 1$ and $lambda = -1$. Thus, I thought that a general solution would be somewhere along the lines of $begin{pmatrix} x'(t) \ y'(t) end{pmatrix} = c_1e^tbegin{pmatrix}frac{1}{2} \ frac{sqrt{3}}{2} end{pmatrix} + c_2e^{-t}begin{pmatrix} -frac{sqrt{3}}{2} \ frac{1}{2} end{pmatrix}$.

I am unsure about how to go from here to finding specific solutions. I'm asked to find two solutions to this system "without performing calculations", and to indicate where the solutions curves start in the plane. Furthermore, I'm asked to find a solution that starts at $begin{pmatrix} 0 \ 5 end{pmatrix}$.

What am I missing here?

$$A^{-1}=A,$$
$$e^{At}=A.
begin{pmatrix}{e^t} & 0\
0 & {e^{-t}}end{pmatrix}
.A^{-1}$$
Final solution is
$$overrightarrow{w}(t) =e^{At}.begin{pmatrix} 0 \ 5 end{pmatrix}
=begin{pmatrix}frac{5 sqrt{3}, {{e}^{t}}}{4}-frac{5 sqrt{3}, {{e}^{-t}}}{4}\
frac{15 {{e}^{t}}}{4}+frac{5 {{e}^{-t}}}{4}end{pmatrix} $$