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 $[A: B]=\begin{bmatrix}1 & -1 & -2 & 6 \\ -1 & 1 & 1 & \mu \\ \lambda & 1 & 1 & 3\end{bmatrix}$
$\sim\begin{bmatrix}1 & -1 & -2 & 6 \\ 0 & 0 & -1 & \mu+6 \\ \lambda+1 & 0 & -1 & 9\end{bmatrix}$
Applying $R _{2} \rightarrow R _{2}+R_{1}, R _{3} \rightarrow R _{3}+ R _{1}$
$\sim\begin{bmatrix}1 & -1 & -2 & 6 \\ 0 & 0 & -1 & \mu+6 \\ \lambda+1 & 0 & 0 & 3-\mu\end{bmatrix}$
Infinite number of solutions for $\lambda=-1$ and $\mu=3.$