The system of equations $x + y + z = 2$,$3x - y + 2z = 6$ and $3x + y + z = -18$ has

The number of $3 \times 3$ matrices $A$,whose entries are either $1$ or $-1$ and for which the system $A\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}$ has exactly three distinct solutions,is

If $\begin{bmatrix} 3 & 1 \\ 4 & 1 \end{bmatrix} X = \begin{bmatrix} 5 & -1 \\ 2 & 3 \end{bmatrix}$,then $X =$

Let $\beta$ be a real number. Consider the matrix $A = \begin{bmatrix} \beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2 \end{bmatrix}$. If $A^7 - (\beta - 1)A^6 - \beta A^5$ is a singular matrix,then the value of $9\beta$ is:

If the system of simultaneous linear equations $3x - 4y + kz + 13 = 0$,$x + 2y - z - 9 = 0$,and $kx - y + 3z + 7 = 0$ has a unique solution $x = \alpha, y = \beta, z = \gamma$ for $k \neq m$ and $2\beta - \gamma = 8$,then $\alpha + m =$

If the system of linear equations $2x + 2ay + az = 0$,$2x + 3by + bz = 0$,and $2x + 4cy + cz = 0$,where $a, b, c \in R$ are non-zero and distinct,has a non-zero solution,then: