Question

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

Answer 1

The correct option is D $0$

The equation $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$has two distinct solutions. It should be noted here that the given equation is linear equation in $3$ variables, which may have no solution, or unique solution or infinitely many solutions.
Hence there does not exist any matrix $A$ such that the given equation has exactly two solutions and consequently number of $3\times 3$ matrices is $0$.