Question

Investigate for what values of $\lambda$, $\mu$ the simultaneous equations $x+y+z=6,x+2y+3z=10,x+2y+\lambda z=\mu$ have no solution.

Answer 1

$A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right]$ and $B=\left[\begin{array}{c}6\\ 10\\ \mu \end{array}\right]$

for no solution , $\left|A\right|=0$ and $A_z\ne 0$

So, $\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=0\Rightarrow \lambda =3$

and $A_z\ne 0\Rightarrow \left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|\ne 0\Rightarrow \mu \ne 10$