Mathematics - Determinant Question with Solution | TestHub

$\begin{array}{rl}& \Delta =\left|\begin{array}{ccc}1& 4& -1\\ 7& 9& \mu \\ 5& 1& 2\end{array}\right|=0\\ & \Rightarrow (18-\mu )-4(14-5\mu )-(7-45)=0\Rightarrow \mu =0\\ & \Delta ={\Delta }_{\mathrm{x}}={\Delta }_{\mathrm{y}}={\Delta }_{\mathrm{z}}=0\text{ (For infinite solution) }\\ & {\Delta }_{\mathrm{x}}=\left|\begin{array}{ccc}\lambda & 4& -1\\ -3& 9& \mu \\ -1& 1& 2\end{array}\right|=0\\ & \lambda (18-\mu )-4(-6+\mu )-1(-3+9)=0\\ & 18\lambda +24-6=0\Rightarrow \lambda =-1\end{array}$