JEE Main · 2021 · 31 Aug 2021 · Shift 2 · Matrices

The system is homogeneous, so it admits non-trivial, hence infinitely many, solutions precisely when the coefficient determinant vanishes. That determinant equals $1-{\cos }^2\alpha -{\cos }^2\beta -{\cos }^2\gamma +2\cos \alpha \cos \beta \cos \gamma$. With $\alpha +\beta +\gamma =2\pi$, $\cos \gamma =\cos (\alpha +\beta )$, and $2\cos \alpha \cos \beta \cos (\alpha +\beta )-{\cos }^2(\alpha +\beta )=\cos (\alpha +\beta )\cos (\alpha -\beta )={\cos }^2\alpha -{\sin }^2\beta$. Substituting, the determinant becomes $1-{\cos }^2\alpha -{\cos }^2\beta +{\cos }^2\alpha -{\sin }^2\beta =0$. The determinant is zero, so the system has infinitely many solutions.