= Solution
For the <homogeneous solution>, expand in the independent <eigenvectors> of $A$. Each coefficient obeys $c_j'=\lambda_jc_j$, so the <complementary function> is
$$
\boxed{\mathbf x_c(t)=\sum_{j=1}^nC_j\mathbf a_j e^{\lambda_jt}.}
$$
If $\mathbf x_p$ is any <particular integral>, subtracting it from any other solution gives a <homogeneous solution>. Hence $\mathbf x=\mathbf x_p+\mathbf x_c$ is the <general solution>. Complex-conjugate modes can be combined to produce real solutions when the forcing is real.
For a real two-by-two matrix, the characteristic polynomial is $\lambda^2-(\operatorname{tr}A)\lambda+\det A$. Under the stated distinct-eigenvalue assumption, all nonzero homogeneous modes are purely oscillatory exactly when the <eigenvalues> are $\pm i\omega$ with $\omega>0$. Their sum and product imply $\operatorname{tr}A=0$, $\det A=\omega^2>0$. Conversely these <trace> and <determinant> conditions force those <eigenvalues>, so there is no exponential growth, decay or Jordan-block secular growth.
For the final specified initial-value problem, direct multiplication gives $A^2=-9I$. The forcing vector $\mathbf b=(2,3i-1)^T$ satisfies $A\mathbf b=3i\mathbf b$, so the forcing is resonant and $t\mathbf b e^{3it}$ is a <particular integral>. The <matrix exponential> is
$$
e^{At}=I\cos3t+\frac A3\sin3t.
$$
Since the resonant <particular integral> vanishes initially, the initial-value solution is
$$
\boxed{\mathbf x(t)=
\begin{pmatrix}\cos3t+\tfrac13\sin3t\\-\tfrac53\sin3t\end{pmatrix}
+t e^{3it}\begin{pmatrix}2\\3i-1\end{pmatrix}.}
$$
This complex solution is appropriate to the complex forcing as written. Taking real parts gives the corresponding solution for the real part of that forcing.
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