linear algebra - spectral radius of a matrix as one element changes

Here's my question --

Let $A$ be an $n times n$ real matrix, and suppose that the spectral radius $rho(A)$ is less than one (spectral radius = max eigenvalue). Let's choose some $1 leq i leq N$ and look at $A_{N,i}$. Namely, let's replace $A_{N,i}$ with some new value, $a$, to give us a new matrix $hat A$. I want to characterize the set $lbrace a : rho(hat A) < 1 rbrace$. It pretty clear that this set is of the form $[0, a_{max})$, but I want to be able to compute $a_{max}$ analytically, given $A$ and $i$. (Also clearly $a_{max} geq A_{N,i}$, since $rho(A) < 1$ by assumption.)

This seems like it should be a fairly easy exercise but I haven't been able to make any useful progress on it.

Thanks!

-h

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