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, $|A| \neq 0$.
$\Rightarrow\left|\begin{array}{lll}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda\end{array}\right| \neq 0$
$\Rightarrow 1(2 \lambda-9)-1(\lambda-3)+1(3-2) \neq 0$
$\Rightarrow 2 \lambda-9-\lambda+3+1 \neq 0$
$\Rightarrow \lambda-5 \neq 0$
$\Rightarrow \lambda \neq 5$
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