A particle moves along a straight line marked with integers. The probability of moving from point $i$ to $i-1$ in one step is $p$, the probability of staying at point $i$ is $q$, and the probability of moving to point $i+1$ is $r$, with $p+q+r=1$. Find the one-step transition probability matrix and the two-step transition probability matrix.

Options:
A. $$
P^{(1)}=\left\{\begin{matrix} {{0,}} & {{j < i-2}} \\ {{p,}} & {{j=i-1}} \\ {{q,}} & {{j=i}} \\ {{r,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\2p^2\:,&j=i-2\\pq\:,&j=i-1\\q^2+pr\:,&j=i\\rq\:,&j=i+1\\r^2\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
B. $$
P^{(1)}=\left\{\begin{matrix} {{p,}} & {{j < i-1}} \\ {{0,}} & {{j=i-1}} \\ {{r,}} & {{j=i}} \\ {{q,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\p^2\:,&j=i-2\\2pq+r\:,&j=i-1\\q^2\:,&j=i\\2pr\:,&j=i+1\\0\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
C. $$
P^{(1)}=\left\{\begin{matrix} {{0,}} & {{j < i-1}} \\ {{p,}} & {{j=i-1}} \\ {{q,}} & {{j=i}} \\ {{r,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\p^2\:,&j=i-2\\2pq\:,&j=i-1\\q^2+2pr\:,&j=i\\2rq\:,&j=i+1\\r^2\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
D. $$
P^{(1)}=\left\{\begin{matrix} {{0,}} & {{j < i-1}} \\ {{q,}} & {{j=i-1}} \\ {{0,}} & {{j=i}} \\ {{p,}} & {{j=i+1}} \\ {{r,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\p^2\:,&j=i-2\\2pq+r\:,&j=i-1\\q^2\:,&j=i\\pr+2r^2\:,&j=i+1\\0\:,&j=i+2\\0\:,&j>i+1\end{cases}$$
E. $$
P^{(1)}=\left\{\begin{matrix} {{0,}} & {{j < i-1}} \\ {{r,}} & {{j=i-1}} \\ {{p,}} & {{j=i}} \\ {{q,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\r^2\:,&j=i-2\\2pq\:,&j=i-1\\q^2\:,&j=i\\2pr+2qr\:,&j=i+1\\p^2\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
F. $$
P^{(1)}=\left\{\begin{matrix} {{0,}} & {{j < i-1}} \\ {{q,}} & {{j=i-1}} \\ {{0,}} & {{j=i}} \\ {{r,}} & {{j=i+1}} \\ {{p,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\p^2\:,&j=i-2\\pq+2r^2\:,&j=i-1\\q^2+pr\:,&j=i\\2qr\:,&j=i+1\\0\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
G. $$
P^{(1)}=\left\{\begin{matrix} {{r,}} & {{j < i-1}} \\ {{q,}} & {{j=i-1}} \\ {{p,}} & {{j=i}} \\ {{0,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\p^2+rq\:,&j=i-2\\0\:,&j=i-1\\q^2\:,&j=i\\2pr\:,&j=i+1\\r^2\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
H. $$
P^{(1)}=\left\{\begin{matrix} {{0,}} & {{j < i-1}} \\ {{p,}} & {{j=i-1}} \\ {{r,}} & {{j=i}} \\ {{q,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-3\\p^2\:,&j=i-2\\pq+rq\:,&j=i-1\\q^2+pr\:,&j=i\\r^2\:,&j=i+1\\0\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
I. $$
P^{(1)}=\left\{\begin{matrix} {{p,}} & {{j < i-1}} \\ {{r,}} & {{j=i-1}} \\ {{q,}} & {{j=i}} \\ {{0,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-2\\p^2\:,&j=i-2\\pq\:,&j=i-1\\2q^2\:,&j=i\\r^2+2pr\:,&j=i+1\\0\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
J. $$
P^{(1)}=\left\{\begin{matrix} {{0,}} & {{j < i-1}} \\ {{q,}} & {{j=i-1}} \\ {{r,}} & {{j=i}} \\ {{p,}} & {{j=i+1}} \\ {{0,}} & {{j > i+1}} \\ \end{matrix} \right.
$$
$$P^{(2)}=\begin{cases}0\:,&j<i-1\\p^2\:,&j=i-2\\2pq\:,&j=i-1\\r^2+2pr\:,&j=i\\q^2\:,&j=i+1\\0\:,&j=i+2\\0\:,&j>i+2\end{cases}$$
Answer: C