Let $S =\{ M = [a_{ij}], a_{ij} \in \{0,1,2\}, 1 \leq i, j \leq 2\}$ be a sample space and $A = \{M \in S : M \text{ is invertible}\}$ be an event. Then $P(A)$ is equal to

Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$,for which $|\frac{z-a}{z+b}|=1$ and $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1$ for some $z \in \mathbb{C}$,where $\omega$ and $\omega^2$ are the roots of $x^2+x+1=0$,is equal to . . . . . .

If $A(\theta)=\begin{bmatrix} i \sin \theta & \cos \theta \\ \cos \theta & i \sin \theta \end{bmatrix}$ is a matrix,where $i=\sqrt{-1}$,then which of the following is not true?

Let $p$ be an odd prime number and $T_{p}$ be the set of $2 \times 2$ matrices defined as:
$T_p = \left\{ A = \begin{bmatrix} a & b \\ c & a \end{bmatrix} : a, b, c \in \{0, 1, \ldots, p-1\} \right\}$
$1.$ The number of matrices $A \in T_p$ such that $A$ is either symmetric or skew-symmetric or both,and $\det(A)$ is divisible by $p$ is:
$(A) (p-1)^2$ $(B) 2(p-1)$ $(C) (p-1)^2+1$ $(D) 2p-1$
$2.$ The number of matrices $A \in T_p$ such that the trace of $A$ is not divisible by $p$ but $\det(A)$ is divisible by $p$ is:
$(A) (p-1)(p^2-p+1)$ $(B) p^3-(p-1)^2$ $(C) (p-1)^2$ $(D) (p-1)(p^2-2)$
$3.$ The number of matrices $A \in T_p$ such that $\det(A)$ is not divisible by $p$ is:
$(A) 2p^2$ $(B) p^3-5p$ $(C) p^3-3p$ $(D) p^3-p^2$

If matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$,then which of the following statements is incorrect?

If $\Delta_{r}=\left|\begin{array}{cc}\frac{1}{3r-2} & \frac{2}{3r-5} \\ 0 & \frac{3}{3r+1}\end{array}\right|$,then $\sum_{r=1}^{33} \Delta_{r}=$