The system of equations:
Question:

The system of equations:

$x+y+z=5$

$x+2 y+3 z=9$

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

has a unique solution, if

(a) $\lambda=5, \mu=13$

(b) $\lambda \neq 5$

(c) $\lambda=5, \mu \neq 13$

(d) $\mu \neq 13$

Solution:

$(\mathrm{b}) \lambda \neq 5$

For a unique solution, $|A| \neq 0$.

$\Rightarrow\left|\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda\end{array}\right| \neq 0$

$\Rightarrow 1(2 \lambda-9)-1(\lambda-3)+1(3-2) \neq 0$

$\Rightarrow 2 \lambda-9-\lambda+3+1 \neq 0$

$\Rightarrow \lambda-5 \neq 0$

$\Rightarrow \lambda \neq 5$