Exercise 3.2.54

If $c=1,R=\left[\begin{array}{cccc}1\hfill & 1\hfill & 2\hfill & 2\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]$ has $x_2,x_3,x_4$ free. If $c\ne 1,R=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ has $x_3,x_4$ free. Special solutions in $N=\left[\begin{array}{ccc}\hfill -1\hfill & \hfill -2\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ (for $c=1$ ) and $N=$ $\left[\begin{array}{cc}\hfill -2& \hfill -2\\ \hfill 0& \hfill 0\\ \hfill 1& \hfill 0\\ \hfill 0& \hfill 1\end{array}\right]$(for $c\ne 1$ ). If $c=1,R=\left[\begin{array}{cc}0\hfill & 1\hfill \\ 0\hfill & 0\hfill \end{array}\right]$and $x_1$ free; if $c=2,R=\left[\begin{array}{cc}1\hfill & -2\hfill \\ 0\hfill & 0\hfill \end{array}\right]$ and $x_2$ free; $R=I$ if $c\ne 1,2.$ Special solutions in $N=\left[\begin{array}{c}1\hfill \\ 0\hfill \end{array}\right](c=1)$ or $N=$ $\left[\begin{array}{c}2\hfill \\ 1\hfill \end{array}\right](c=2)$or $N=2$ by 0 empty matrix.