Question

Let $\lambda \in R.$ The system of linear equations
$2x_1-4x_2+\lambda x_3=1$
$x_1-6x_2+x_3=2$
$\lambda x_1-10x_2+4x_3=3$
is inconsistent for:

• A. exactly two values of $\lambda$.
• B. exactly one negative value of $\lambda$.
• C. every value of $\lambda$.
• D. exactly one positive value of $\lambda$.

Answer 1

The correct option is B exactly one negative value of $\lambda$.

$D=\left|\begin{array}{ccc}2& -4& \lambda \\ 1& -6& 1\\ \lambda & -10& 4\end{array}\right|$
$=2(3\lambda +2)(\lambda -3)$
$D_1=-2(\lambda -3)$
$D_2=-2(\lambda +1)(\lambda -3)$
$D_3=-2(\lambda -3)$
When $\lambda =3,$ then
$D=D_1=D_2=D_3=0$
$\Rightarrow$ Infinite many solution.

When $\lambda =\frac{-2}{3},$ then $D_1,D_2,D_3$ none of them is zero so equations are inconsistent.
$\therefore \lambda =\frac{-2}{3}$