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 are $1$ and four of them are $0$

Consider the system of linear equations $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, then which of the following is/are correct?

• A. The total number of possible matrices in $A$ is $12$.
• B. The number of matrices $A$ in set $A$ for which the system of linear equations has a unique solution is $6$.
• C. The number of matrices $A$ in set $A$ for which the system of linear equations has a unique solution is $4$.
• D. The number of matrices $A$ in set $A$ for which the system of linear equations is inconsistent is more than $2$.

Answer 1

Answer: (A), (B), (D)

The correct options are
A The total number of possible matrices in $A$ is $12$.
B The number of matrices $A$ in set $A$ for which the system of linear equations has a unique solution is $6$.
D The number of matrices $A$ in set $A$ for which the system of linear equations is inconsistent is more than $2$.

Let the matrix
$A=\left[\begin{array}{ccc}a& b& c\\ l& m& n\\ p& q& r\end{array}\right]$
Given:
Five entries as $1$ and remaining four entries as $0$.
Since, the matrix is symmetric, it must have even number of zeros for $i\ne j$.

Therefore, there are two cases:
Case $\left(1\right)$
Two entries in diagonal are zero.
We can select two places from three (in diagonal) in
${}^3{C}_2$ ways
Now,
We have to select elements for upper triangle.
For upper triangle, we have three places of which one entry is ${}^{\prime }{0}^{\prime }$ and two are ${}^{\prime }{1}^{\prime }$.
Therefore, one place from three can be selected in ${}^3{C}_1$ ways.
Hence, the number of matrices
$={}^3{C}_2\times {}^3{C}_1=9$.

Case $\left(2\right)$
If all the entries in the principal diagonal are $1$, we have two ${}^{\prime }{0}^{\prime }$ and one ${}^{\prime }{1}^{\prime }$ in upper triangle.
Hence, the number of matrices $=3$.

Therefore, the total number of matrices in $A$ is $3+9=12$

Now, for the sytstem of equation to have unique solution $|A|\ne 0$
Consider the matrix
$A=\left[\begin{array}{ccc}0& a& b\\ a& 0& c\\ b& c& 1\end{array}\right]$
$|A|=a^2$
So,
$|A|\ne 0$ if either $b=0$ or $c=0$
$\therefore$ number of matrices $=2$

Now, Consider the matrix $A=\left[\begin{array}{ccc}0& a& b\\ a& 1& c\\ b& c& 0\end{array}\right]$
$|A|=b^2$
So,
$|A|\ne 0$ if either $a=0$ or $c=0$
$\therefore$ number of matrices $=2$

Now, Consider the matrix $A=\left[\begin{array}{ccc}1& a& b\\ a& 0& c\\ b& c& 0\end{array}\right]$
Clearly, $|A|\ne 0$ if either $a=0$ or $b=0$
$\therefore$ number of matrices $=2$

Now, Consider the matrix $A=\left[\begin{array}{ccc}1& a& b\\ a& 1& c\\ b& c& 1\end{array}\right]$
Clearly,
$a=b=0,c=1\Rightarrow |A|=0$
$a=c=0,b=1\Rightarrow |A|=0$
$b=c=0,a=1\Rightarrow |A|=0$
$\therefore$ number of matrices in this case $=0.$

Hence, total number of matrices having unique solutions is $6$.


The six matrices $A$ for which $|A|=0$ are:
$A=\left[\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 1& 1& 1\end{array}\right]\Rightarrow \text{inconsistent}$

$A=\left[\begin{array}{ccc}0& 1& 0\\ 1& 1& 1\\ 0& 1& 0\end{array}\right]\Rightarrow \text{inconsistent}$

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 0& 0\\ 1& 0& 0\end{array}\right]\Rightarrow \text{infinite solutions}$

$A=\left[\begin{array}{ccc}1& 1& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right]\Rightarrow \text{inconsistent}$

$A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 1& 0& 1\end{array}\right]\Rightarrow \text{inconsistent}$

$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 1& 1\end{array}\right]\Rightarrow \text{inconsistent}$