Exercise 3.5.27

$B$ and $C$ (checkers and chess) both have rank 2 if $p\ne 0$. Row 1 and 2 are a basis for the row space of $C,B^{T}y=0$ has 6 special solutions with -1 and 1 separated by a zero; $N\left(C^T\right)$ has $(-1,0,0,0,0,0,0,1)$ and $(0,-1,0,0,0,0,1,0)$ and columns $3,4,5,6$ of $I;N(C)$ is a challenge : one vector in $N(C)$ is $(1,0,\dots ,0,-1)$.