Question

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

• A. 0
• B. $2^9 -1$
• C. 168
• D. 2

Answer 1

Answer: (A)

Since, $A\begin {bmatrix}x \\y \\z \end {bmatrix}-\begin {bmatrix}1 \\0 \\0 \end {bmatrix}$ is linear equation in three variables and that could have only unique, no solution or infinitely many solution.
$\therefore $ It is not possible to have two solutions.
Hence, number of matrices $A$ is zero