Question

Let $A$ be the set of all $3\times 3$ symmetric matrices all 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 $3$ ways; 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

$A=A_6,A_8,A_{10},A_{11}$