If $A=\begin{bmatrix} 1 & 5 & 3 \\ 2 & 4 & 0 \\ 3 & -1 & -5 \end{bmatrix}$,$B=\begin{bmatrix} -1 \\ -2 \\ 4 \end{bmatrix}$ and $[x \ y \ z] A^{T}=B^{T}$,then $x+y+z=$

The system of equations $4x + y + 2z = 5$,$x - 5y + 3z = 10$,and $9x - 3y + 7z = 20$ has

If the set of equations $x+2y+3z=6$,$x+3y+5z=9$,and $2x+5y+az=b$ has a unique solution,then:

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ and $B = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} \neq \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ such that $AB = B$ and $a + d = 2021$. Then the value of $ad - bc$ is equal to ...... .

The solution of the equation $\begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \\ 0 & -1 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}$ is $(x, y, z) = $

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: