Complex eigenvalues

Eigenvector for $\lambda=2$. $u = (1,1,1)$, giving the solution $e^{2t}u$.

Complex pair. For $\lambda = 1+i$, find a complex eigenvector, then take real and imaginary parts of $e^{(1+i)t}v$. The real solutions are of the form
$$X(t) = e^t\begin{pmatrix} a \\ b \\ c \end{pmatrix}\cos t + e^t\begin{pmatrix} d \\ e \\ f \end{pmatrix}\sin t,$$
where the coefficients are determined by substituting into $X' = AX$.