Discrete time impulse response

Question

The following question is from a System Theory exam whitout answers or solutions:

Which of the following discrete-time state-space model (A,B,C) of the form

$x(t+1)=Ax(t)+Bu(t), \quad y(t)=Cx(t), \quad t \in N$

With $A$ in Jordan frm has its impulse response given by

$ h(t)=\left\{ \begin{array}{ll} 1, \quad t=4 \\ 0, \quad t \neq 4 \end{array} \right. $

$A) \left[ \begin{array}{c|c} A & B \\ \hline C & \end{array} \right] = \left[ \begin{array}{cccc|c} 0&1&0&0&0 \\ 0&0&1&0&0 \\ 0&0&0&1&0 \\ 0&0&0&0&1 \\ \hline 1&0&0&0 \end{array} \right] $

$B) \left[ \begin{array}{c|c} A & B \\ \hline C & \end{array} \right] = \left[ \begin{array}{cccc|c} 0&1&0&0&1 \\ 0&0&1&0&0 \\ 0&0&0&1&0 \\ 0&0&0&0&0 \\ \hline 0&0&0&1 \end{array} \right] $

$C) \left[ \begin{array}{c|c} A & B \\ \hline C & \end{array} \right] = \left[ \begin{array}{cccc|c} 1&1&0&0&0 \\ 0&1&1&0&0 \\ 0&0&1&1&0 \\ 0&0&0&1&1 \\ \hline 0&1&0&0 \end{array} \right] $

$D) \left[ \begin{array}{c|c} A & B \\ \hline C & \end{array} \right] = \left[ \begin{array}{cccc|c} 1&1&0&0&0 \\ 0&1&1&0&0 \\ 0&0&1&1&0 \\ 0&0&0&1&1 \\ \hline 0&1&0&0 \end{array} \right] $

$E)$ None of the above

I just found out how to calculate this but since I've gone through all the trouble of typing the question, I might ass well post it anyway. Maybe it helps someone else.

The answer can be found using $CA^{t-1}B$

So we get $CA^3B=1$ and $CA^{\neq 3}B=0$

We first try option $A$ which meets the criteria and thus is the only correct result.

Answer 1

I just found out how to calculate this but since I've gone through all the trouble of typing the question, I might ass well post it anyway. Maybe it helps someone else.

The answer can be found using $CA^{t−1}B$

So we get $CA^3B=1$ and $CA^{≠3}B=0$

We first try option A which meets the criteria and thus is the only correct result.