The number of $3 \times 3$ matrices $A$ whose entries are either $0$ or $1$ and for which the system $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ has exactly two distinct solutions, is

The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system A x y z = 1 0 0 has exactly two distinct solutions, is $A = [a_{ij}], \quad a_{ij} \in \{0,1\}, \quad 3\times 3$ $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ $\text{To find: number of } 3\times 3\ \{0,1\}\text{-matrices } A \text{ for which}$ $\text{the system has exactly two distinct solutions}$ $A\mathbf{x} = \mathbf{b}, \qquad \mathbf{b} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ $\text{Number of solutions is always } 0,\ 1, \text{ or } \infty$ $A\mathbf{x}_1 = \mathbf{b}, \quad A\mathbf{x}_2 = \mathbf{b} \Rightarrow A(\mathbf{x}_1 - \mathbf{x}_2) = \mathbf{0}$ $A\big(\mathbf{x}_1 + t(\mathbf{x}_1 - \mathbf{x}_2)\big) = \mathbf{b} \quad \forall\, t \in \mathbb{R}$ $\mathbf{x}_1 - \mathbf{x}_2 \neq \mathbf{0} \Rightarrow \text{infinitely many solutions}$ $\Rightarrow \text{exactly two distinct solutions is impossible}$ $\text{Solution set} = \varnothing,\ \text{a point},\ \text{a line},\ \text{or a plane}$ $\text{Number of such matrices } A = 0$ $\boxed{\text{Answer: (1) } 0}$