Stability of time stepping in PDE

Question

I am currently trying to perform a stability assessment on solving a parabolic PDE of the form $$ \frac{\partial u}{\partial t} = g(x,t) \frac{\partial^2 u}{\partial x^2} + u $$ I am using Crank-Nicolson in space and just forward Euler in time. I know that to assess stability, we write the PDE in the form $$ u^{n+1} = A \cdot u^n + f $$ and check to see if the eigenvalues of $A$ are less than 1. Now, if the matrix $A$ varies in time, would this assessment need to be performed at each time step? I am doing this in order to bound my time step.