Is it possible to determine the value of a matrix element given the dominant eigenvalue and all other elements? I'm working on a population dynamics model and have a matrix of vital rates representing the survival and fecundity of different life stages of the animal which are set out in a 6 x 6 matrix:
\begin{pmatrix}
0 & 0 & 0 & 0 & 3 & 4\\ 
0.5 & 0 & 0 & 0 & 0 & 0 \\ 
0 & 0.6 & 0 & 0 & 0 & 0\\ 
0 & 0 & 0.7 & 0 & 0 & 0\\ 
0 & 0 & 0 & 0.8 & 0 & 0\\ 
0 & 0 & 0 & 0 & 0.9 & x
\end{pmatrix}
I also have a vector of initial population sizes:
\begin{pmatrix}
10\\ 
20\\ 
25\\ 
25\\ 
70\\ 
80
\end{pmatrix}
I already have an estimate for the x element in the matrix which allows me to calculate the population growth rate or dominant eigenvalue lambda. I was wondering if it's possible to calculate what value should x take if I want a lambda of 1 which is of biological relevance to the growth rate. 

I don't know if negative values of $x$ make sense for your model (perhaps not), but I got $x\simeq-0.22$. For that $x$, there are only two real eigenvalues, one of which is (very nearly) $1$; the other one is about $-0.85$.
