If the system of equations
$x+y+az=b$
$2x+5y+2z=6$
$x+2y+3z=3$
has infinitely many solutions,then $2a+3b$ is equal to $...........$.

If $\begin{bmatrix} 2 & 1 & -1 \\ -1 & 2 & 1 \\ 1 & -1 & 2 \end{bmatrix} \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} 4 \\ 0 \\ -2 \end{bmatrix}$, then the distance of point $P(a, b, c)$ from the plane $2x + y + 2z = 10$ is

If $\begin{bmatrix} 3 & 1 \\ 4 & 1 \end{bmatrix} X = \begin{bmatrix} 5 & -1 \\ 2 & 3 \end{bmatrix}$,then $X =$

If the system $\begin{bmatrix} 2 & 8 \\ 3 & 7 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = k \begin{bmatrix} a \\ b \end{bmatrix}$ has a non-trivial solution, then the positive value of $k$ and a solution of the system for that value of $k$ are:

Let $k_1$ and $k_2$ be the maximum and minimum values of $k$ for which the system of equations $x + ky = 1$,$kx + y = 2$,and $x + y = k$ are consistent. Then $k_1^2 + k_2^2$ is equal to:

For the system of linear equations $a x+y+z=1$,$x+a y+z=1$,$x+y+a z=\beta$,which one of the following statements is $NOT$ correct?