Let $n$ be the number obtained on rolling a fair die. If the probability that the system of equations
$x-ny+z=6$
$x+(n-2)y+(n+1)z=8$
$(n-1)y+z=1$
has a unique solution is $\frac{k}{6}$, then the sum of $k$ and all possible values of $n$ is:

Let $a, b, c$ be positive real numbers. The following system of equations in $x, y, z$:
$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$
$\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
$-\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$
has:

Consider the following system of equations in matrix form $\begin{bmatrix} 1 \\ 2 \\ \lambda \end{bmatrix} \begin{bmatrix} 1 & 2 & \lambda \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = 0$. Then which one of the following statements is true?

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:

Examine the consistency of the system of equations: $x+2y=2$ and $2x+3y=3$.

Let the system of equations $x+2y+3z=5$,$2x+3y+z=9$,and $4x+3y+\lambda z=\mu$ have an infinite number of solutions. Then $\lambda+2\mu$ is equal to: