Question

Under what condition does the above system of equations have unique solutions?

• A. $\lambda =5$ and $\mu =9$
• B. $\lambda \ne 5$ and $\mu =7$ only
• C. $\lambda \ne 5$ and $\mu$ has any real value
• D. $\lambda$ has any real value and $\mu \ne 9$

Answer 1

The correct option is A $\lambda =5$ and $\mu =9$

$2x+3y+5z=9$,

$7x+3y-2z=8$,

$2x+3y+\lambda z=\mu$

Making the augmented matrix,

$\left[\begin{array}{cccc}2& 3& 5& 9\\ 7& 3& -2& 8\\ 2& 3& \lambda & \mu \end{array}\right]$ $R_3\to R_3-R_1$

$\left[\begin{array}{cccc}2& 3& 5& 9\\ 7& 3& -2& 8\\ 0& 0& \lambda -5& \mu -9\end{array}\right]$

Now for a unique solution, we should have $\lambda \ne 5$

and $\mu$ can have any real value.

Hence, C is correct.