Question

The system of linear equations
$\lambda x+y+z=3$
$x-y-2z=6$
$-x+y+z=\mu$ has

• A. Infinite number of solutions for $\lambda \ne -1$ and all $\mu$
• B. Infinite number of solutions for $\lambda =-1$ and $\mu =3$
• C. No solution for $\lambda \ne -1$
• D. Unique solution for $\lambda =-1$ and $\mu =3$

Answer 1

Answer: (B)

Augmented matrix $[A:B]=\left[\begin{array}{cccc}1& -1& -2& 6\\ -1& 1& 1& \mu \\ \lambda & 1& 1& 3\end{array}\right]$
$\sim \left[\begin{array}{cccc}1& -1& -2& 6\\ 0& 0& -1& \mu +6\\ \lambda +1& 0& -1& 9\end{array}\right]$
Applying $R_2\to R_2+R_1,R_3\to R_3+R_1$
$\sim \left[\begin{array}{cccc}1& -1& -2& 6\\ 0& 0& -1& \mu +6\\ \lambda +1& 0& 0& 3-\mu \end{array}\right]$
Infinite number of solutions for $\lambda =-1$ and $\mu =3.$