If the system of linear equations $2\mathrm{x}+3\mathrm{y}-\mathrm{z}=-2$, $\mathrm{x}+\mathrm{y}+\mathrm{z}=4$, $\mathrm{x}-\mathrm{y}+\left|\mathrm{\lambda }\right|\mathrm{z}=4\mathrm{\lambda }-4$ where $\mathrm{\lambda }\in \mathrm{ℝ}$, has no solution, then

Augumented matrix is

$$\begin{gathered} \left[\begin{array}{cccc}1& 1& 1& 4\\ 2& 3& -1& -2\\ 1& -1& \left|\lambda \right|& 4\lambda -4\end{array}\right]~ \\ {R}_2\to R_2-2R_1,R_3\to R_3-R_1 \\ ~\left[\begin{array}{cccc}1& 1& 1& 4\\ 0& 1& -3& -10\\ 0& -2& \left|\lambda \right|-1& 4\lambda -8\end{array}\right] \\ {R}_3\to R_3+2R_2 \\ ~\left[\begin{array}{cccc}1& 1& 1& 4\\ 0& 1& -3& -10\\ 0& 0& \left|\lambda \right|-7& 4\lambda -28\end{array}\right]\begin{array}{} \\ \\ \end{array} \end{gathered}$$

Given system of equations has no solution 

$$\begin{gathered} \Rightarrow \left|\lambda \right|-7=0 and 4\lambda -28\ne 0 \\ \therefore \lambda =-7 \end{gathered}$$