The number of solutions of the equations $x + y - z = 0$,$3x - y - z = 0$,and $x - 3y + z = 0$ is

If the system of equations $x+y+z=1$,$x+2y+4z=k$ and $x+4y+10z=k^2$ is consistent,then $k$ is equal to

Statement $1$: If the system of equations $x + ky + 3z = 0, 3x + ky - 2z = 0, 2x + 3y - 4z = 0$ has a nontrivial solution,then the value of $k$ is $\frac{31}{2}$.
Statement $2$: $A$ system of three homogeneous equations in three variables has a nontrivial solution if the determinant of the coefficient matrix is zero.

Let $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right]$,$B=\left[B_1, B_2, B_3\right]$,where $B_1, B_2, B_3$ are column matrices,and $AB_1=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$,$AB_2=\left[\begin{array}{l}2 \\ 3 \\ 0\end{array}\right]$,$AB_3=\left[\begin{array}{l}3 \\ 2 \\ 1\end{array}\right]$. If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$,then $\alpha^3+\beta^3$ is equal to

If the system of linear equations $3x + y + \beta z = 3$,$2x + \alpha y - z = -3$,and $x + 2y + z = 4$ has infinitely many solutions,then the value of $22\beta - 9\alpha$ is:

Let $S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z = 1$,$x + \lambda y + z = 1$,and $x + y + \lambda z = 1$ is inconsistent. Then,$\sum_{\lambda \in S} (|\lambda|^2 + |\lambda|)$ is equal to