Consider a time varying non-negative matrix $A(t)$ and its spectral radius $\rho(A(t))$ where $t$ denotes the time. If $A(t)$ changes over time with each time a random element in $A(t)$ is being increased by a random value, will the spectral radius be monotonously increasing over time as well?
In other words, can I say that:
$\rho(A(1))\leq\rho(A(2))\ldots\leq\rho(A(N))$
If yes, how do I proof this?
If no, why?