Question

Let A be the set of all 3 $\times$ 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.

The number of matrices A in A for which the system of linear equations $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ is inconsistent is

• A. 0
• B. more than 2
• C. 2
• D. 1

Answer 1

The correct option is B more than 2

Let $x$ be the number of 1's on the main diagonal and $y$ be the number of 1's above the main diagonal, then
$x+2y=5$
$\Rightarrow x=1,y=2orx=3,y=1$.
When $x=1$, the main diagonal can be chosen in 3 ways, and the elements above the main diagonal in 3 ways. Therefore, there are 9 such matrices.These are
$A_1=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right]$, $A_2=\left[\begin{array}{ccc}0& 1& 1\\ 1& 1& 0\\ 1& 0& 0\end{array}\right]$
$A_3=\left[\begin{array}{ccc}0& 1& 1\\ 1& 0& 0\\ 1& 0& 1\end{array}\right]$, $A_4=\left[\begin{array}{ccc}1& 0& 1\\ 0& 0& 1\\ 1& 1& 0\end{array}\right]$
$A_5=\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 1\\ 1& 1& 0\end{array}\right]$, $A_6=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]$
$A_7=\left[\begin{array}{ccc}1& 1& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$, $A_8=\left[\begin{array}{ccc}0& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right]$
$A_9=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 1& 1\end{array}\right]$
When $x=3$, the main diagonal can be chosen in 3 way; and the element above the main diagonal in 3 ways. Therefore, there are 3 such matrices.
$A_{10}=\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right]$, $A_{11}=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]$,
$A_{12}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 1& 1\end{array}\right]$
Let $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, and note that
$A_{1}X=B$ and $A_{12}X=B$ have infinite number of solutions.
$AX=B$ has no solution when
$A=A_6,A_8,A_{10},A_{11}$