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. 

The matrix you gave has characteristic polynomial
$$ p(\lambda ) = \lambda ^6 - x \lambda^5 - 0.504 \lambda + 0.504 x - 0.6048 $$
A necessary condition for $1$ to be an eigenvalue of the matrix is that $p(1) = 0$. This gives us an equation for $x$:
$$0 = 1 - x -0.504 + 0.504x - 0.6048$$
Solving for $x$ we get
$$x = -0.219355 $$
Not sure if that value makes sense biologically, as all the other entries are positive, but if you want an eigenvalue of $1$ for the matrix, you'll have to go with that $x$.
