A two-state system updates by xk+1=Axk\mathbf{x}_{k+1} = A\mathbf{x}_kxk+1=Axk, starting from x0\mathbf{x}_0x0:
A=[4123],x0=[52]A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}, \qquad \mathbf{x}_0 = \begin{bmatrix} 5 \\ 2 \end{bmatrix}A=[4213],x0=[52]
Use the eigendecomposition A=PDP−1A = PDP^{-1}A=PDP−1, which makes A3=PD3P−1A^3 = PD^3P^{-1}A3=PD3P−1, to advance the system three steps.
What is x3=A3x0\mathbf{x}_3 = A^3\mathbf{x}_0x3=A3x0?
Select all that apply.
(322, 94)(322,\; 94)(322,94)
(157, −218)(157,\; -218)(157,−218)
(66, 48)(66,\; 48)(66,48)
(508, 484)(508,\; 484)(508,484)