The system of equations $4x + y - 2z = 0$,$x - 2y + z = 0$,and $x + y - z = 0$ has

For what value of $k$ does the following system of equations possess a non-trivial solution?
$x + ky + 3z = 0$
$3x + ky - 2z = 0$
$2x + 3y - 4z = 0$

For the system $x-y+z=4, 2x+y-3z=0, x+y+z=2$,the values of $x, y, z$ respectively are given by

The system of equations $x+y+z=6$,$x+2y+5z=9$,$x+5y+\lambda z=\mu$ has no solution if

Let $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$ such that $AX = B$,then $X =$

Let $S$ be the set of values of $\lambda$,for which the system of equations
$6 \lambda x - 3 y + 3 z = 4 \lambda^2$
$2 x + 6 \lambda y + 4 z = 1$
$3 x + 2 y + 3 \lambda z = \lambda$
has no solution. Then $12 \sum_{\lambda \in S} |\lambda|$ is equal to $...........$.