Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A)$ the sum of diagonal entries of $A$. Assume that $A^2 = I$.
Statement-$1$: If $A \neq I$ and $A \neq -I$,then $\det(A) = -1$.
Statement-$2$: If $A \neq I$ and $A \neq -I$,then $tr(A) \neq 0$.

Let $A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix}$. If $B = I - {}^{3}C_{1}(\operatorname{adj} A) + {}^{3}C_{2}(\operatorname{adj} A)^{2} - {}^{3}C_{3}(\operatorname{adj} A)^{3}$,then the sum of all elements of the matrix $B$ is

If $\omega$ is an imaginary cube root of unity,then the value of $\left[\begin{array}{ccc}1 & \omega^{2} & 1-\omega^{4} \\ \omega & 1 & 1+\omega^{5} \\ 1 & \omega & \omega^{2}\end{array}\right]$ is

Let $A = [a_{ij}]$,where $a_{ij} \in \mathbb{Z} \cap [0, 4]$ and $1 \leq i, j \leq 2$. The number of matrices $A$ such that the sum of all entries is a prime number $p \in (2, 13)$ is $........$.

$A$ determinant is chosen at random from the set of all determinants of order $2$ with elements $0$ or $1$ only. The probability that the value of the chosen determinant is positive is:

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 $2$ $(C) 2$ $(D) 1$