If the system of equations $3x - 2y + z = 0$, $\lambda x - 14y + 15z = 0$, and $x + 2y - 3z = 0$ has a solution other than $x = y = z = 0$, then $\lambda = $

Let $A=\begin{bmatrix} 0 \\ -6 \\ 8 \end{bmatrix}$,$B=\begin{bmatrix} 3 & 5 & -7 \\ 0 & -1 & 8 \\ 6 & -1 & 0 \end{bmatrix}$ and $X=\begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $D=[\alpha, \beta, \gamma]^{T}$ is the solution of $X^{T} B^{T}=A^{T}$,then $D^{T} A=$

The existence of the unique solution of the system $x + y + z = \lambda$,$5x - y + \mu z = 10$,and $2x + 3y - z = 6$ depends on

Solve the system of linear equations using the matrix method: $x-y+z=4$,$2x+y-3z=0$,and $x+y+z=2$.

Let $A = \begin{bmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9 \end{bmatrix}$ and $B = \begin{bmatrix} a \\ -b \\ -c \end{bmatrix}$. If $A$ and $[A: B]$ have the same rank, then:

Let $[\lambda]$ be the greatest integer less than or equal to $\lambda$. The set of all values of $\lambda$ for which the system of linear equations $x+y+z=4$,$3x+2y+5z=3$,$9x+4y+(28+[\lambda])z=[\lambda]$ has a solution is: