Question

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

• A. $0$
• B. $2^9-1$
• C. $168$
• D. $2$

Answer 1

The correct option is A $0$

$A_{3\times 3}\cdot B_{3\times 1}=C_{3\times 1}$ $\left(1\right)$
A is a matrix whose entries are either $0$ or $1$
$\therefore$ AB will be a $3\times 1$ matrix with entries x, y, z. Thus, equation $\left(1\right)$ represents the equation of three planes. We know that three planes can not intersect at two distinct points. Hence the number of solutions is zero.