Question

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system ${\text{A}}\left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  0 \\
  0
\end{array}} \right]$ has exactly two distinct solutions, is:
$\left( a \right)0$
$\left( b \right){2^9} - 1$
$\left( c \right)168$
$\left( d \right)2$

Answer 1

Hint: In this particular question use the concept that three different planes can only intersect at most one point they cannot intersect more than one point so use these concepts to reach the solution of the question.

Complete step-by-step solution:
Given data:
${\text{A}}\left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  0 \\
  0
\end{array}} \right]$................... (1), where A is a $3 \times 3$ matrices A whose entries are either 0 or 1.
Now we have to find out the number of such A matrices so that the system has exactly two distinct solutions.
Let, $A = \left[ {\begin{array}{*{20}{c}}
  {{a_1}}&{{b_1}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}} \\
  {{a_3}}&{{b_3}}&{{c_3}}
\end{array}} \right]$
Now substitute this value from equation (1) we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
  {{a_1}}&{{b_1}}&{{c_1}} \\
  {{a_2}}&{{b_2}}&{{c_2}} \\
  {{a_3}}&{{b_3}}&{{c_3}}
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
  x \\
  y \\
  z
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
  1 \\
  0 \\
  0
\end{array}} \right]$
$ \Rightarrow {a_1}x + {b_1}y + {c_1}z = 1$................ (2)
$ \Rightarrow {a_2}x + {b_2}y + {c_2}z = 0$............... (3)
$ \Rightarrow {a_3}x + {b_3}y + {c_3}z = 0$................ (4)
Now equation (2), (3), and (4) represent equations of linear planes in 3 dimensional.
Now as we know that linear planes can intersect at most one point otherwise they cannot be linear planes.
So equation (2), (3), and (4) can intersect at most one point.
So the plane has only one solution.
So the system has only one solution.
It is not possible to have more than one distinct solution.
Hence the number of such A matrices so that the system has exactly two distinct solutions is zero.
So this is the required answer.
Hence option (a) is the correct answer.

Note: Whenever we have such types of questions the key concept we have to remember is that linear planes always intersect at one point and non – linear planes such as circle, hyperbola, parabola, etc. can intersect more than one point.