Question

Given $2x-y+2z=1;x-2y+z=-4\text{and}x+y+\lambda z=4.$
Then the value of $\lambda$ for which the given system of equations has no solution is

• A. 1
• B. 2
• C. -1
• D. -2

Answer 1

The correct option is A 1

The given system of equations can be represented in matrix form as:
AX = B, where
$A=\left[\begin{array}{ccc}2& -1& 2\\ 1& -2& 1\\ 1& 1& \lambda \end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]andB=\left[\begin{array}{c}1\\ -4\\ 4\end{array}\right]$
For no solution, $|A|=\Delta =0\text{and any one of}{\Delta }_1,{\Delta }_2,or{\Delta }_3\text{should be non-zero.}$
$\Delta =\left|\begin{array}{ccc}2& -1& 2\\ 1& -2& 1\\ 1& 1& \lambda \end{array}\right|=0$
$\Rightarrow \lambda =1$
${\Delta }_3=\left|\begin{array}{ccc}2& -1& 1\\ 1& -2& -4\\ 1& 1& 4\end{array}\right|=3\ne 0$
So $\lambda =1$