Consider the simultaneous linear equations $\beta x + \alpha y - z = -1$,$3x - \beta y + \alpha z = 0$,and $\alpha x + \beta y + z = 1$. In the usual notation used in Cramer's rule,given that $\frac{\Delta_1}{\Delta} = -1$,$\frac{\Delta_2}{\Delta} = 1$,and $\frac{\Delta_3}{\Delta} = 2$,then $(\alpha, \beta) = $

The positive value of $a$ for which the system of linear homogeneous equations $x+ay+z=0$,$ax+2y-z=0$,and $2x+3y+z=0$ has non-trivial solutions is

If $A$ and $B$ are the two real values of $k$ for which the system of equations $x+2y+z=1$,$x+3y+4z=k$,and $x+5y+10z=k^2$ is consistent,then $A+B=$

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:

The system of linear equations $3x - 2y - kz = 10$,$2x - 4y - 2z = 6$,and $x + 2y - z = 5m$ is inconsistent if

Let $A$ be a $3 \times 3$ real matrix such that $A\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = 2\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}$,$A\begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix} = 4\begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}$,and $A\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = 2\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}$. Then,the system $(A-3I)\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ has