I have been trying to develop to solve a state space model in discrete time. I have the following equation for my state space: $$\dot{x} = Ax + Bu, $$
I am developping the equation, but I do not understand why I don't have the right form as equations 7 and 8 in this website (link): $$ \dot{x} = \frac{dx}{dt} = f(x, t) \\ \implies \frac{x_{n} - x_{n-1}}{h} = f(x_n, t_n) \\ \implies \frac{x_{n} - x_{n-1}}{h} = Ax_n + Bu_n \\ \implies x_n - x_{n-1} = hAx_n + hBu_n \\ \implies (I - hA)x_n = x_{n-1} + hBu_n \\ \implies \boxed{x_n = (I-hA)^{-1}x_{n-1} - (I-hA)^{-1}hBu_n} $$$$ \dot{x} = \frac{dx}{dt} = f(x, t) \\ \implies \frac{x_{n} - x_{n-1}}{h} = f(x_n, t_n) \\ \implies \frac{x_{n} - x_{n-1}}{h} = Ax_n + Bu_n \\ \implies x_n - x_{n-1} = hAx_n + hBu_n \\ \implies (I - hA)x_n = x_{n-1} + hBu_n \\ \implies \boxed{x_n = (I-hA)^{-1}x_{n-1} + (I-hA)^{-1}hBu_n} $$
I should obtain $ x_n = (I-hA)^{-1}x_{n-1} + hABu_{n-1} $ according to equation 8 here.