$$X'=\begin{bmatrix} 0 & 0 & a\\0 & b & 0 \\a &0 &0\\ \end{bmatrix}X$$
I calculated the eigenvalues to be $\lambda=a,-a,b$ with eigenvectors $$v_a=\begin{bmatrix} 1&0&1 \end{bmatrix}\\$$ $$v_{-a}=\begin{bmatrix} -1&0&1 \end{bmatrix}\\$$ $$v_b=\begin{bmatrix} 0&1&0 \end{bmatrix}\\$$
I formed my solution as
$$X(t)=c_1e^{at}\begin{bmatrix} 1&0&1 \end{bmatrix}^T+c_2e^{-at}\begin{bmatrix} -1&0&1 \end{bmatrix}^T+c_3e^{bt}\begin{bmatrix} 0&1&0 \end{bmatrix}^T$$
Is this the correct approach to solving 3x3 matrix differential systems?
I am also confused on how to sketch the phase portrait of this system in the ab plane.