Question

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

Answer 1

A non-homogenous system has unique solution if $D\ne 0$ and infinite solutions if $D=D_1=D_2=D_3=0$

Now $D=\left|\begin{array}{ccc}1& 1& -2\\ 2& -3& 1\\ 1& -5& 4\end{array}\right|$

Hence $D_1,D_2,D_3$ should all be zero

$\Rightarrow D_1=\left|\begin{array}{ccc}0& 1& -2\\ 0& -3& 1\\ \lambda & -5& 4\end{array}\right|=0$

$\Rightarrow 0(-3-5)-1(0-\lambda )-2(0+3\lambda )=0$

$\Rightarrow \lambda -6\lambda =-5\lambda =0$

$\therefore \lambda =0$