\begin{equation*}
A-\lambda I=\begin{pmatrix}
1-\lambda & 0 & 0\\
2 & 1-\lambda & -2\\
3 & 2 & 1-\lambda
\end{pmatrix}
\end{equation*}
\begin{equation*}
\det(A-\lambda I)=(1-\lambda)[(1-\lambda)^2+4]=0
\end{equation*}
\begin{equation*}
\end{equation*}
\begin{equation*}
\lambda_1=1~~~~\lambda_2=1+2i~~~~\lambda_2=1-2i
\end{equation*}
When $\lambda_1=1$
\begin{equation*}
\begin{pmatrix}
0 & 0 & 0\\
2 & 0 & -2\\
3 & 2 & 0
\end{pmatrix}
\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix}=0
\end{equation*}
let x1 = t
\begin{equation*}
\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix}=t
\begin{pmatrix}
1\\
-3/2\\
1
\end{pmatrix}
\end{equation*}
When $\lambda_2=1+2i$
\begin{equation*}
\begin{pmatrix}
-2i & 0 & 0\\
2 & -2i & -2\\
3 & 2 & -2i
\end{pmatrix}
\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix}=0
\end{equation*}
let x2 = t
\begin{equation*}
\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix}=t
\begin{pmatrix}
0\\
i\\
1
\end{pmatrix}
\end{equation*}
When $\lambda_3=1-2i$
\begin{equation*}
\begin{pmatrix}
2i & 0 & 0\\
2 & 2i & -2\\
3 & 2 & 2i
\end{pmatrix}
\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix}=0
\end{equation*}
let x2 = t
\begin{equation*}
\begin{pmatrix}
x_1\\
x_2\\
x_3
\end{pmatrix}=t
\begin{pmatrix}
0\\
1\\
i
\end{pmatrix}
\end{equation*}
\begin{equation*}
e^{(1+2i)t}\begin{pmatrix}
0\\
i\\
1
\end{pmatrix}=\begin{pmatrix}
0\\
i\\
1
\end{pmatrix}e^te^{2it}=\begin{pmatrix}
0\\
icos2t-sin2t\\
cos2t+isin2t
\end{pmatrix}e^t=e^t\begin{pmatrix}
0\\
-sin2t\\
cos2t
\end{pmatrix}+ie^t\begin{pmatrix}
0\\
cos2t\\
sin2t
\end{pmatrix}
\end{equation*}
\begin{equation*}
e^{(1-2i)t}\begin{pmatrix}
0\\
1\\
i
\end{pmatrix}=\begin{pmatrix}
0\\
1\\
i
\end{pmatrix}e^te^{-2it}=\begin{pmatrix}
0\\
cos(-2t)+isin(-2t)\\
-icos(-2t)-sin(-2t)
\end{pmatrix}e^t=e^t\begin{pmatrix}
0\\
cos(-2t)\\
-sin(-2t)
\end{pmatrix}+ie^t\begin{pmatrix}
0\\
sin(-2t)\\
cos(-2t)
\end{pmatrix}
\end{equation*}
so, the real-valued function is
\begin{equation*}
x(t)=C_1e^t\begin{pmatrix}
1\\
-3/2\\
1
\end{pmatrix}+C_2e^t\begin{pmatrix}
0\\
-sin2t\\
cos2t
\end{pmatrix}+C_3e^t\begin{pmatrix}
0\\
cos(-2t)\\
sin(2t)
\end{pmatrix}
\end{equation*}
