If $A = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix}$ and square matrix $B$ satisfy $AB = 8I$, then the value of $|B|$ is:

The real value of $a$ for which the system of equations $x+ay=0, y+az=0, z+ax=0$ has infinitely many solutions, is :

Let $A=\begin{bmatrix}1 & \sin\theta & 1\\-\sin\theta & 1 & \sin\theta\\-1 & -\sin\theta & 1\end{bmatrix}$, where $\theta\in[0,2\pi]$ then which of the following statements are correct?

A. $|A|=2+2\sin^2\theta$

B. Maximum value of $|A|$ is 1.

C. Maximum value of $|A|$ is –1.

D. $|A|\in[2,4]$

Choose the correct answer from the options given below:

If the matrix $\left[\begin{array}{rrr}0 & -1 & 3 x \\ 1 & y & -5 \\ -6 & 5 & 0\end{array}\right]$ is skew-symmetric, then the value of $5 x-y$ is:

For some constant 'k', if the system of linear equations

$2x - y + 3z = 1$

$x - 2y + z = 3$

$kx + y - z = 0$

has a unique solution, then

If A is a square matrix and I is an identity matrix of same order such that $A^2 = A$, then $(I + A)^3 - 8I$ is equal to

Let $A = [a_{ij}]_{3 \times 3}$ be a matrix, defined by $a_{ij} = \begin{cases} 2i+3j & , i < j \\6 &, i=j\\ 3i-2j & , i > j \end{cases}$. The number of elements in A which are greater than 6, is