The existence of the unique solution of the system of equations $2x + y + z = \beta$,$10x - y + \alpha z = 10$ and $4x + 3y - z = 6$ depends on

If the system of linear equations $8x + y + 4z = -2$,$x + y + z = 0$,and $\lambda x - 3y = \mu$ has infinitely many solutions,then the distance of the point $\left(\lambda, \mu, -\frac{1}{2}\right)$ from the plane $8x + y + 4z + 2 = 0$ is:

Let the system of equations: $2x + 3y + 5z = 9$,$7x + 3y - 2z = 8$,$12x + 3y - (4 + \lambda)z = 16 - \mu$ have infinitely many solutions. Then the radius of the circle centred at $(\lambda, \mu)$ and touching the line $4x = 3y$ is

For a matrix $A=\begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$,if $U_{1}, U_{2}$,and $U_{3}$ are $3 \times 1$ column matrices satisfying $A U_{1}=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$,$A U_{2}=\begin{bmatrix} 2 \\ 3 \\ 0 \end{bmatrix}$,$A U_{3}=\begin{bmatrix} 2 \\ 3 \\ 1 \end{bmatrix}$,and $U$ is a $3 \times 3$ matrix whose columns are $U_{1}, U_{2}$,and $U_{3}$,then the sum of the elements of $U^{-1}$ is:

For what value of $\lambda$,the system of equations $x + y + z = 6$,$x + 2y + 3z = 10$,and $x + 2y + \lambda z = 12$ is inconsistent? $\lambda = $ ........

Consider the following system of equations: $x+2y-3z=a$,$2x+6y-11z=b$,and $x-2y+7z=c$,where $a, b$,and $c$ are real constants. Then the system of equations: