Mathematics - Determinant Question with Solution | TestHub

Here, we will use Cramer's Rule according to which, $D=0$ and, $D_1\ne 0$ if the system of linear equations is inconsistent. $D=\left|\begin{array}{ccc}2& -4& \lambda \\ 1& -6& 1\\ \lambda & -10& 4\end{array}\right|=2\left(-24+10\right)+4\left(4-\lambda \right)+\lambda \left(-10+6\lambda \right)$

$=2\left(3{\lambda }^2-7\lambda -6\right)=2\left(\lambda -3\right)\left(3\lambda +2\right)$

$D=0\Rightarrow \lambda =3,-\frac{2}{3}$

$D_1=\left|\begin{array}{ccc}1& -4& \lambda \\ 2& -6& 1\\ 3& -10& 4\end{array}\right|=1\left(-24+10\right)+4\left(8-3\right)+\lambda \left(-20+18\right)=2\left(3-\lambda \right)$

For $\lambda =-\frac{2}{3},{\mathrm{D}}_1\ne 0$