Solve the system of linear equations using the matrix method: $x-y+2z=7$,$3x+4y-5z=-5$,$2x-y+3z=12$.

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be a $2 \times 2$ real matrix with $\det(A) = 1$. If the equation $\det(A - \lambda I_2) = 0$ has imaginary roots (where $I_2$ is the identity matrix of order $2$),then:

The system of linear equations $x + y + z = 2$,$2x + y - z = 3$,and $3x + 2y + kz = 4$ has a unique solution if

If the system of simultaneous linear equations $x+y-z=6$,$4x+y+z=2$,and $x+ky+z=-8$ has a unique solution $x=2$,$y=\beta$,$z=\gamma$,then the value of $k$ satisfies which of the following quadratic equations?

Let $a, \lambda, \mu \in \mathbb{R}$. Consider the system of linear equations:
$a x + 2 y = \lambda$
$3 x - 2 y = \mu$
Which of the following statement$(s)$ is(are) correct?
$(A)$ If $a = -3$,then the system has infinitely many solutions for all values of $\lambda$ and $\mu$.
$(B)$ If $a \neq -3$,then the system has a unique solution for all values of $\lambda$ and $\mu$.
$(C)$ If $\lambda + \mu = 0$,then the system has infinitely many solutions for $a = -3$.
$(D)$ If $\lambda + \mu \neq 0$,then the system has no solution for $a = -3$.

If the system of equations $x + 2y + 3z = 4$,$x + py + 2z = 3$,and $x + 4y + \mu z = 3$ has an infinite number of solutions,then: