JEE Main · 2020 · 04 Sep 2020 · Shift 2 · Matrices

For infinitely many solutions the coefficient determinant must vanish: $\left|\begin{array}{ccc}1& 1& 1\\ 2& 4& -1\\ 3& 2& \lambda \end{array}\right|=0$ gives $2\lambda -9=0$, so $\lambda =\frac{9}{2}$. Consistency then forces the determinant with the constants placed in the third column to vanish as well, $\left|\begin{array}{ccc}1& 1& 2\\ 2& 4& 6\\ 3& 2& \mu \end{array}\right|=0$, which gives $\mu =5$. These values satisfy $2\lambda +\mu =9+5=14$.