Question:
The existence of the unique solution of the system of equations:
$x+y+z=\lambda$
$5 x-y+\mu z=10$
$2 x+3 y-z=6$
depends on
(a) $\mu$ only
(b) $\lambda$ only
(c) $\lambda$ and $\mu$ both
(d) neither $\lambda$ nor $\mu$
Solution:
(a) $\mu$ only
For a unique solution, $|A| \neq 0$
$\Rightarrow\left|\begin{array}{ccc}1 & 1 & 1 \\ 5 & -1 & \mu \\ 2 & 3 & -1\end{array}\right| \neq 0$
$\Rightarrow 1(1-3 \mu)-1(-5-2 \mu)+1(15+2) \neq 0$
$\Rightarrow 1-3 \mu+5+2 \mu+17 \neq 0$
$\Rightarrow-\mu+23 \neq 0$
$\Rightarrow \mu \neq 23$
So, existence of a unique solution depends only on $\mu$.