Fast computation for matrices to compute a sequence of states from a discrete-time, time-varying state space model

Hello

I have the following linear discrete-time, time-varying state space model

No i want to compute a sequence of states from a sequence of inputs with the following equation:

My question is, how can i compute the augemented matrices and in a fast way (e.g. without loops, built-in function, ... ) for big N (e.g. N>100)?

The matrices and are already computed for all and they are saved in 3-dimensional arrays and .