The number of $3 x 3$ matrices A whose entries are either $0$ or $1$ for which the system $A\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}0\\ 0\\ 0\end{array}\right]$ has exact y two distinct
solutions is

Let $A=\left[\begin{array}{l}{a}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{c}_1\\ a_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{c}_2\\ a_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{b}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{c}_3\end{array}\right]$. The system becomes $a_{1}x+b_{1}y+c_{1}z=0,a_{2}x+b_{2}y+c_{2}z=0$ and $a_{3}x+b_{3}y+c_{3}z=0$ representing $3$ planes.  Three planes meet at 
(i) exactly one point, . 
(ii) no point or 
(iii) infinite number of points. They cannot meet at exactly two points for any $A$.