Question

The number of $3\times$ 3 matrices A whose entries are either 0 or 1 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

• A. 0
• B. $2^9-1$
• C. 168
• D. 2

Answer 1

Answer: (A)

0

Let $A=\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]$
where these elements are 0 or 1.
Given $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$
$\left[\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$
$\left[\begin{array}{c}{a}_{1}x+a_{2}y+a_{3}z\\ b_{1}x+b_{2}y+b_{3}z\\ c_{1}x+c_{2}y+c_{3}z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$
$\Rightarrow a_{1}x+a_{2}y+a_{3}z=0$
$b_{1}x+b_{2}y+b_{3}z=0$
$c_{1}x+c_{2}y+c_{3}z=0$
$\Rightarrow b_1=b_2=b_3=c_1=c_2=c_3=0$
$a_{1}x+a_{2}y+a_{3}z=0$
This cannot have exactly two distinct solutions , i.e. not possible .
Hence, there is no such A matrix.