JEE Main · 2026 · 04 Apr 2026 · Shift 2 · Matrices

The coefficient determinant is $\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& \lambda \end{array}\right|=\lambda -5$, and infinitely many solutions require it to vanish, so $\lambda =5$. Subtracting the first equation from the second gives $y+2z=4$, and from the third gives $2y+4z=\mu -5$. For these to be consistent, $\mu -5$ must equal $2(4)=8$, so $\mu =13$. Hence $\lambda +\mu =5+13=18$.