Key Concepts

This problem involves solving a system of homogeneous linear equations represented by a matrix equation $PX=0$, along with an additional constraint $x^2 + y^2 + z^2 = 1$.

1. Homogeneous System of Linear Equations: For a system $PX=0$ where $P$ is an $n \times n$ matrix, the nature of solutions depends on the determinant of $P$:
   • If $\det(P) \neq 0$, the system has only the trivial solution $X = \mathbf{0}$ (i.e., $x=y=z=0$).
   • If $\det(P) = 0$, the system has non-trivial solutions (inf