For $3 \times 3$ matrices $M$ and $N$,which of the following statement$(s)$ is (are) $NOT$ correct?

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 $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ 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 $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ is inconsistent,is
$(A) 0$ $(B)$ more than $2$ $(C) 2$ $(D) 1$

Let $A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$ and $B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ be two $2 \times 1$ matrices with real entries such that $A = XB$,where $X = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}$ and $k \in R$. If $a_1^2 + a_2^2 = \frac{2}{3}(b_1^2 + b_2^2)$ and $(k^2 + 1)b_2^2 \neq -2b_1b_2$,then the value of $k$ is ....... .

Let $A^{-1} = \begin{bmatrix} 1 & 2017 & 2 \\ 1 & 2017 & 4 \\ 1 & 2018 & 8 \end{bmatrix}$. Then,$|2A| - |2A^{-1}|$ is equal to

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix}$ where each of $a$,$b$,and $c$ is either $\omega$ or $\omega^2$. Then,the number of distinct matrices in the set $S$ is

Let $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP - 2I|$ is equal to