JEE Main · 2026 · 02 Apr 2026 · Shift 1 · Matrices

A system has no solution when the coefficient determinant vanishes while the equations are inconsistent. Here $\Delta =\left|\begin{array}{ccc}1& 2& 1\\ 2& 1& \alpha \\ 8& 4& \beta \end{array}\right|=1(\beta -4\alpha )-2(2\beta -8\alpha )+1(8-8)=12\alpha -3\beta$. Setting $\Delta =0$ gives $\beta =4\alpha$. With $\beta =4\alpha$ the left side of the third equation is exactly $4$ times that of the second (since $4E_2$ reproduces $8x+4y+4\alpha z$), yet the right side $18\ne 4\cdot 5=20$, so the system is inconsistent and has no solution. Therefore $\frac{\beta }{\alpha }=4$.