Question

If the system of equations $x+y+\lambda z=0,\lambda x–2y–z=0$ and $3x–y+z=1$ is inconsistent, then $\lambda$ is equal to

• A. $-2$
• B. $3$
• C. $\frac{1}{2}$
• D. $2$

Answer 1

Answer: (B), (D)

$2$

$\Delta =\left|\begin{array}{ccc}1& 1& \lambda \\ \lambda & -2& -1\\ 3& -1& 1\end{array}\right|$
$=–3–\lambda –3–{\lambda }^2+6\lambda =–({\lambda }^2–5\lambda +6)=–(\lambda –2)(\lambda –3)$
${\Delta }_1=\left|\begin{array}{ccc}0& 1& \lambda \\ 0& -2& -1\\ 1& -1& 1\end{array}\right|=(2\lambda –1),{\Delta }_2=\left|\begin{array}{ccc}1& 0& \lambda \\ \lambda & 0& -1\\ 3& 1& 1\end{array}\right|=1+{\lambda }^2$
${\Delta }_3=\left|\begin{array}{ccc}1& 1& 0\\ \lambda & -2& 0\\ 3& -1& 1\end{array}\right|=–(\lambda +2)$