Let $X = \left\{\begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \mathbb{R} \right\}$. Define $f: X \rightarrow \mathbb{R}$ by $f(A) = \operatorname{det}(A), \forall A \in X$. Then, $f$ is

Let $A$ be a non-singular matrix of order $3$. If $\operatorname{det}(\operatorname{adj}(2 \operatorname{adj}((\operatorname{det} A) A))) = 3^{-13} \cdot 2^{-10}$ and $\operatorname{det}(\operatorname{adj}(2A)) = 2^m \cdot 3^n$,then $|3m + 2n|$ is equal to:

If matrix $A = [a_{ij}]_{3 \times 3}$ and $B = [b_{ij}]_{3 \times 3}$,where $a_{ij} + a_{ji} = 0$ and $b_{ij} - b_{ji} = 0$ for all $i, j$,then $A^4B^3$ is:

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:

Let $A = \begin{bmatrix} 2 & b & 1 \\ b & b^2+1 & b \\ 1 & b & 2 \end{bmatrix}$ where $b > 0$. Then the minimum value of $\frac{\det(A)}{b}$ is

If $A=\left[\begin{array}{cc}i & 0 \\ 0 & -i\end{array}\right]$,$B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$ and $C=\left[\begin{array}{cc}0 & i \\ i & 0\end{array}\right]$,then which of the following is true?