Let $A = \left| \begin{array}{cc} 2 & e^{i \pi} \\ -1 & i^{2012} \end{array} \right|$,$C = \left. \frac{d}{dx} \left( \frac{1}{x} \right) \right|_{x=1}$,and $D = \int_{e^2}^{1} \frac{dx}{x}$. If the sum of two roots of the equation $Ax^3 + Bx^2 + Cx - D = 0$ is equal to zero,then $B$ is equal to:

For $i=1, 2, 3$ and $j=1, 2, 3$. If $a_i^2+b_i^2+c_i^2=1$,$a_i a_j+b_i b_j+c_i c_j=0$,$\forall i \neq j$ and $A=\begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix}$,then $\det(AA^T)=$

Matrix $A$ is such that ${A^2} = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,${A^n} = $

Let $A, B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ identity matrix such that $AB-C$ is a non-singular matrix. Let $D=(AB-C)^{-1}$. Then, consider the following statements.
Statement $I$: $\operatorname{det}(BA)=\operatorname{det}(BA-C) \operatorname{det}(BDA)$
Statement $II$: $ABD=DAB$
Which of the above statements is (are) true?

Let $A$ be a $3 \times 3$ matrix such that $A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}$ and $A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}$. If $\det(A) = 1$ and the matrix $A$ satisfies $A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}$, then $\det(\text{adj}(A^2 + A))$ is equal to:

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 $........$.