JEE Main · 2026 · 21 Jan 2026 · Shift 2 · Matrices

A linear system fails to have a solution when the coefficient determinant vanishes while the equations remain inconsistent. Here $\Delta =\left|\begin{array}{ccc}3& 1& 4\\ 2& \alpha & -1\\ 1& 2& 1\end{array}\right|=3(\alpha +2)-1(2+1)+4(4-\alpha )=19-\alpha$. Setting $\Delta =0$ gives $\alpha =19$. For this value the coefficient matrix has rank $2$ while the augmented matrix has rank $3$, so the system is inconsistent and has no solution. The value of $\alpha$ is $19$.