Question

Given

$\begin{array}{c}2x-y+2z=2\\ x-2y+z=-4\\ x+y+\lambda z=4\end{array}$

then the value of $\lambda$ such that the given system of equations has no solution, is ?

• A. 3
• B. 1
• C. 0
• D. -3

Answer 1

The correct option is B 1

Given system of equations $AX=B$
where $A=\left[\begin{array}{ccc}2& -1& 2\\ 1& -2& 1\\ 1& 1& \lambda \end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}2\\ -4\\ 4\end{array}\right]$
For no solution,
$D=0$
$\left|\begin{array}{ccc}2& -1& 2\\ 1& -2& 1\\ 1& 1& \lambda \end{array}\right|=0$
$\Rightarrow -3\lambda +3=0$
$\Rightarrow \lambda =1$
Now, $D_1=\left|\begin{array}{ccc}2& -1& 2\\ -4& -2& 1\\ 4& 1& \lambda \end{array}\right|$
$\Rightarrow D_1=-6\ne 0$
Hence, for $\lambda =1$ at least one $D_1,D_2,D_3$ is 0.