The number of $3\times 3$  matrices $A$  whose entries are either $0or1$  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

Since, $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$  is linear equation in three variables and that could have only unique, no solution or infinitely many solutions.

$\therefore$    It is the possible to have two solutions

Hence, number of matrices $A$ is zero.