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Let $\Omega$ 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 $0$ .
$1.$ The number of matrices in $\Omega$ is
$(A)$ $12$ $(B)$ $6$ $(C)$ $9$ $(D)$ $3$
$2.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations
$A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has a unique solution, is
$(A)$ less than $4$
$(B)$ at least $4$ but less than $7$
$(C)$ at least $7$ but less than $10$
$(D)$ at least $10$
$3.$ The number of matrices $A$ in $\Omega$ for which the system of linear equations
$A\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ is inconsistent, is
$(A)$ $0$ $(B)$ more than $2$ $(C)$ $2$ $(D)$ $1$
$(A,B,B)$
$(A,D,C)$
$(A,D,C)$
$(D,B,A)$
Solution
$1.$ Case $I$ All three diagonal elements are $1$
Number of matrices $={ }^3 C _1=3$
Case $II$ Two diagonal elements are zero and one element is one No. of matrices = ${ }^3 C_1 \cdot{ }^3 C_1=9$
So, Total matrices $=3+9=12$
$2.$ $\left[\begin{array}{lll} 0 & a & b \\ a & 0 & c \\ b & c & 1 \end{array}\right]$
Either $b=0$ or $c=0 \Rightarrow|A| \neq 0$
$\Rightarrow$ two matrices
$A=\left[\begin{array}{lll} 0 & a & b \\ a & 1 & c \\ b & c & 0 \end{array}\right]$
Either $a=0$ or $c=0 \Rightarrow|A| \neq 0$
$\Rightarrow$ two matrices
$A=\left[\begin{array}{lll} 1 & a & b \\ a & 0 & c \\ b & c & 0 \end{array}\right]$
Either $a=0$ or $b=0 \Rightarrow|A| \neq 0$
$\Rightarrow$ two matrices
$A=\left[\begin{array}{lll} 1 & a & b \\ a & 1 & c \\ b & c & 1 \end{array}\right]$
$a=b=0 \Rightarrow|A|=0$
$a=c=0 \Rightarrow|A|=0$
$b=c=0 \Rightarrow|A|=0$
Therefore, there will be only six matrices.
