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

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$2^{9} - 1$

$168$

$0$

$2$

answer is A.

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Detailed Solution

Let $A = [\begin{array}{lll} a_{1} b_{1} c_{1} \\ a_{2} b_{2} c_{2} \\ a_{3} b_{3} c_{3} \end{array}]$ . 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 $A$ .