Eigenvalues · Hard problem

A fish population's juveniles $j_k$ and adults $a_k$ evolve each year by $j_{k+1} = 1.5a_k$ (each adult produces $1.5$ juveniles on average) and $a_{k+1} = 0.4j_k + (1-h)a_k$, where $h$ is the fraction of adults harvested each year. The population is exactly sustainable in the long run (neither growing nor shrinking) exactly when $1$ is an eigenvalue of the transition matrix $A = \begin{pmatrix}0 & 1.5\\0.4 & 1-h\end{pmatrix}$. Find the harvesting rate $h$ that makes this true.