Eigenvalues · Challenge problem

For a real number $c$ let $A = \begin{pmatrix}0 & 1\\-c & \tfrac32\end{pmatrix}$. The iterates $A^k\mathbf x$ tend to $\mathbf 0$ as $k \to \infty$ for every starting vector $\mathbf x$ exactly when $c$ lies in an open interval $(a, b)$. Find $a + b$.