The sum of three numbers is $6$. Thrice the third number when added to the first number gives $7$. On adding three times the first number to the sum of the second and third numbers,we get $12$. The product of these numbers is:

Consider the system of equations: $x + y - az = 1$; $2x + ay + z = 1$; $ax + y - z = 2$. Which of the following statements is correct?

If $A$ is a matrix such that $\left[\begin{array}{ll} 2 & 1 \\ 3 & 2 \end{array}\right] A \left[\begin{array}{ll} 1 & 1 \end{array}\right] = \left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right]$, then $A$ is equal to

If $x=\alpha, y=\beta, z=\gamma$ is the solution of the system of equations $2x+3y+z=-1$, $3x+y+z=4$, and $x-3y-2z=1$, then the value of $\beta$ is:

If the system of equations $x + y + z = 5$, $x + 2y + 3z = 9$, $x + 3y + \lambda z = \mu$ has infinitely many solutions, then the value of $\lambda + \mu$ is:

For the system of linear equations:
$x - 2y = 1, x - y + kz = -2, ky + 4z = 6, k \in R$
Consider the following statements:
$(A)$ The system has a unique solution if $k \neq 2, k \neq -2$.
$(B)$ The system has a unique solution if $k = -2$.
$(C)$ The system has a unique solution if $k = 2$.
$(D)$ The system has no solution if $k = 2$.
$(E)$ The system has an infinite number of solutions if $k \neq -2$.
Which of the following statements are correct?