The value of $\lambda$ for which the system of equations $2x - y - z = 12,$ $x - 2y + z = -4,$ and $x + y + \lambda z = 4$ has no solution is

The system of simultaneous linear equations $x-2y+3z=4$, $3x+y-2z=7$, and $2x+3y+z=6$ has

If the system of linear equations $x_1 + 2x_2 + 3x_3 = 6$,$x_1 + 3x_2 + 5x_3 = 9$,and $2x_1 + 5x_2 + ax_3 = b$ is consistent and has an infinite number of solutions,then:

If $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is a solution of the system of equations $AX = B$, where $\text{adj } A = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & 5 \\ 1 & -2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 4 \\ 0 \\ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to:

If the values of $x, y$ and $z$ which satisfy the equations $2x - 3y + 2z + 15 = 0$,$3x + y - z + 2 = 0$ and $x - 3y - 3z + 8 = 0$ simultaneously are $\alpha, \beta$ and $\gamma$ respectively,then:

Under which of the following condition$(s)$ does the system of equations $\begin{bmatrix} 1 & 2 & 4 \\ 2 & 1 & 2 \\ 1 & 2 & a-4 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 4 \\ a \end{bmatrix}$ possess a unique solution?