Question

For what value of $\lambda$, the system of equations
$x+y+z=6$
$x+2y+3z=10$
$x+2y+\lambda z=12$
is inconsistent ?

• A. $\lambda =1$
• B. $\lambda =2$
• C. $\lambda =-2$
• D. $\lambda =3$

Answer 1

The correct option is D $\lambda =3$

Given system of equations $AX=B$
where $A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],B=\left[\begin{array}{c}6\\ 10\\ \mu \end{array}\right]$
For system to be inconsistent,
$D=0$
$\Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=0$
$\Rightarrow 2(\lambda -3)-(\lambda -3)=0$
$\Rightarrow \lambda =3$
One of the $D_1,D_2$ and $D_3$ is non-zero.