If $A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & -2 \\ 3 & -4 \\ 2 & 4 \end{bmatrix}$ then product AB is :

If $A = \begin{bmatrix} 1 & a \\ b & 2 \end{bmatrix}$ and $B = \begin{bmatrix} a & 1 \\ 2 & -3b \end{bmatrix}$ such that $\lvert AB\rvert = \lvert 2B\rvert$ and $ab \neq -\dfrac{2}{3}$ then:

A. $\lvert A\rvert = 2$

B. $\lvert A\rvert = 4$

C. $\lvert B\rvert = 2$

D. $\lvert B\rvert = 4$

Choose the correct answer from the options given below:

If $A$ is a skew-symmetric matrix and $n$ is an odd positive integer, then $A^n$ is :

If $A=\begin{bmatrix} 0 & -\tan\alpha/2 \\ \tan\alpha/2 & 0 \end{bmatrix}$, then the value of $\begin{bmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{bmatrix}$ for all $\alpha \in \mathbb{R}$(set of real numbers), is

(I is identity matrix of order 2)

The system of equations

$x - 3y - 8z = -10$

$2x + 5y + \lambda z = 13$

$3x + y - 4z = 0$

has infinite number of solutions if the value of $\lambda$ is equal to:

If matrix A is of order $2 \times 3$ and B of order $3 \times 2$, then

If $f(x) = \begin{vmatrix} 0 & x-1 & x-2 \\ x+1 & 0 & x-3 \\ x+2 & x+3 & 0 \end{vmatrix}$, then the value of $f(0)$ is equal to: