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Representing product and sum updates of values with matrix difference equations [closed]

A vector v updates all of its elements simultaneously in discrete time steps. At each time step, each element of v may change in one of the following ways: It may remain the same. (ex: v1 = v1) It ...
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Solving a linear recurrence relation with variable coefficients.

I have the following recurrence relation: \begin{equation} A[n]=f_A[n-1] A[n-1] + f_B[n-1]B[n-1], \\ B[n]=g_A[n-1] A[n-1] + g_B[n-1]B[n-1], \end{equation} where ...