$x + ky – z = 0,3x – ky – z = 0$ and $x – 3y + z = 0$ has non-zero solution for $k =$

Let $\lambda \in R .$ The system of linear equations

$2 x_{1}-4 x_{2}+\lambda x_{3}=1$

$x_{1}-6 x_{2}+x_{3}=2$

$\lambda x_{1}-10 x_{2}+4 x_{3}=3$  is inconsistent for 

$\left| {\begin{array}{*{20}{c}}0&a&{ – b}\\{ – a}&0&c\\b&{ – c}&0\end{array}} \right| = $

If the points $(2k, k), (k, 2k)$ and $(k, k)$ with $k > 0$ enclose a triangle of area $18$ square unit then centroid of triangle is equal to

Statement $1$ : If the system of equations $x + ky + 3z = 0, 3x+ ky – 2z = 0, 2x + 3y – 4z = 0$ has a nontrivial solution, then the value of $k$ is $\frac{31}{2}$

Statement $2$ : A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.