JEE Main · 2022 · 25 Jul 2022 · Shift 2 · Matrices

For no solution, the system must be inconsistent. Using the first two equations, we can eliminate variables and find a relation involving $z$. From equations 1 and 2, we get a specific value for the linear combination. For the third equation to be inconsistent with this, the coefficient of $z$ must equal a specific value while the constant term differs. Solving $2x-3y+5z=9$ and $x+3y-z=-18$ gives $3x+4z=-9$. Multiplying equation 2 by 3: $3x+9y-3z=-54$. From equation 3: $3x-y+({\lambda }^2-|\lambda |)z=16$. For inconsistency, we need ${\lambda }^2-|\lambda |=4$. This gives $|\lambda |(|\lambda |-1)=4$. Solving: $|\lambda {|}^2-|\lambda |-4=0$, so $|\lambda |=\frac{1+\sqrt{17}}{2}$. This gives 2 real values: $\lambda =\pm \frac{1+\sqrt{17}}{2}$.