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

Question

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) 0 (B) 29 -1 (C) 168 (D) 2

Tardigrade Admin · Accepted Answer

Since, A[x y z ]-[1 0 0 ] is linear equation in three variables and that could have only unique, no solution or infinitely many solution. ∴ It is not possible to have two solutions. Hence, number of matrices A is zero

Q. The number of $3 \times 3$ matrices $A$ whose entries are either $0$ or $1$ and for which the system $A ⎣ ⎡ x y z ⎦ ⎤ - ⎣ ⎡ 100 ⎦ ⎤$ has exactly two distinct solutions, is

0

$2^{9} - 1$

168

2

Since, $A ⎣ ⎡ x y z ⎦ ⎤ - ⎣ ⎡ 100 ⎦ ⎤$ is linear equation in three variables and that could have only unique, no solution or infinitely many solution.
$∴$ It is not possible to have two solutions.
Hence, number of matrices $A$ is zero

Q. The number of 3×3 matrices A whose entries are either 0 or 1 and for which the system A⎣⎡​xyz​⎦⎤​−⎣⎡​100​⎦⎤​ has exactly two distinct solutions, is

0

29−1

168

2

Since, A⎣⎡​xyz​⎦⎤​−⎣⎡​100​⎦⎤​ is linear equation in three variables and that could have only unique, no solution or infinitely many solution. ∴ It is not possible to have two solutions. Hence, number of matrices A is zero

Q. The number of $3 \times 3$ matrices $A$ whose entries are either $0$ or $1$ and for which the system $A ⎣ ⎡ x y z ⎦ ⎤ - ⎣ ⎡ 100 ⎦ ⎤$ has exactly two distinct solutions, is

$2^{9} - 1$

Since, $A ⎣ ⎡ x y z ⎦ ⎤ - ⎣ ⎡ 100 ⎦ ⎤$ is linear equation in three variables and that could have only unique, no solution or infinitely many solution.
$∴$ It is not possible to have two solutions.
Hence, number of matrices $A$ is zero