  We show that there exists no algorithm that decides for any bilinear system
$(B,v)$ if the growth rate of $(B,v)$ is $1$. This answers a question of Bui
who showed that if the coefficients are positive the growth rate is computable
(i.e., there is an algorithm that outputs the sequence of digits of the growth
rate of $(B,v)$). Our proof is based on a reduction of the computation of the
joint spectral radius of a set of matrices to the computation of the growth
rate of a bilinear system. We also use our reduction to deduce that there
exists no algorithm that approximates the growth rate of a bilinear system with
relative accuracy $\varepsilon$ in time polynomial in the size of the system
and of $\varepsilon$. Our two results hold even if all the coefficients are
nonnegative rationals.
