Question

Find $\lambda$ for which the system of equations $x+y-2z=0$, $2x-3y+z=0,x-5y+4z=\lambda$ is consistent.

Answer 1

Augmented Matrix is
$\left[\begin{array}{cccc}1& 1& -2& 0\\ 2& -3& 1& 0\\ 1& -5& 4& \lambda \end{array}\right]$
applying $R_3\to R_3-R_1$ and $R_2\to R_2-2R_1$ gives
$\left[\begin{array}{cccc}1& 1& -2& 0\\ 0& -5& 5& 0\\ 0& -6& 6& \lambda \end{array}\right]$
taking $5$ common from $R_2$ and $6$ from $R_3$ gives
$\left[\begin{array}{cccc}1& 1& -2& 0\\ 0& -1& 1& 0\\ 0& -1& 1& \lambda /6\end{array}\right]$
Given, system of eqautions is consistent.
$\therefore \lambda =0$