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

The system is consistent when it has at least one solution. Compute the determinant:

$\Delta =\left|\begin{array}{ccc}-k& 3& -14\\ -15& 4& -k\\ -4& 1& 3\end{array}\right|=121-k^2$

For $k\ne \pm 11$, $\Delta \ne 0$, so the system has a unique solution (consistent).

For $k=11$ or $k=-11$, we need to check if the system is consistent by computing ${\Delta }_z$ (determinant with the constant column replacing the z-column). In both cases, ${\Delta }_z\ne 0$, which means the system is inconsistent when $k=\pm 11$.

Therefore, the system is consistent for all $k\in \mathrm{R}-\{-11,11\}$.