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 number of solutions of the following system of linear homogeneous equations $x-y+z=0$,$x+2y-z=0$,and $2x+y+3z=0$ is

Given $A = \begin{bmatrix} 1 & 3 \\ 2 & 2 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If $A - \lambda I$ is a singular matrix,then:

Let the system of linear equations $x+y+\alpha z=2$,$3x+y+z=4$,and $x+2z=1$ have a unique solution $(x^{*}, y^{*}, z^{*})$. If $(\alpha, x^{*}), (y^{*}, \alpha)$ and $(x^{*}, -y^{*})$ are collinear points,then the sum of absolute values of all possible values of $\alpha$ is

If the system of equations $x+ky+3z=-2$,$4x+3y+kz=14$,and $2x+y+2z=3$ can be solved by the matrix inversion method,then:

Consider the system of linear equations:
$-x+y+2z=0$
$3x-ay+5z=1$
$2x-2y-az=7$
Let $S_{1}$ be the set of all $a \in \mathbb{R}$ for which the system is inconsistent and $S_{2}$ be the set of all $a \in \mathbb{R}$ for which the system has infinitely many solutions. If $n(S_{1})$ and $n(S_{2})$ denote the number of elements in $S_{1}$ and $S_{2}$ respectively,then: