JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Matrices

The determinant of the coefficient matrix is so the system has a unique solution precisely when $k\ne 2$ and $k\ne -2$. For $k=2$ the equations are $x-2y=1$, $x-y+2z=-2$ and $y+2z=3$; subtracting the first from the second gives $y+2z=-3$, which contradicts the third, so no solution exists. For $k=-2$ the same subtraction gives $y-2z=-3$, which is the third equation $-y+2z=3$ itself, so the system has infinitely many solutions. The two correct assertions are therefore that the system has a unique solution when $k\ne 2$ and $k\ne -2$, and that it has no solution when $k=2$.