JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Matrices

For the system to have no solution, the determinant of coefficients must be zero:


Expanding gives $|\lambda |=7$, so $\lambda =\pm 7$.

To determine which value gives no solution, eliminate $y$ from equations (2) and (3):
From (2): $x+y+z=4$
From (3): $x-y+|\lambda |z=4\lambda -4$

Adding: $2x+(|\lambda |+1)z=4\lambda$, or $x+\frac{|\lambda |+1}{2}z=2\lambda$.

From (2)-(1): $x+4z=14$.

For inconsistency, these two equations must be parallel but not identical. When $\lambda =-7$, we get $|\lambda |=7$, and the system becomes inconsistent.