This problem tests your understanding of homogeneous systems of linear equations and properties of determinants. For a system of homogeneous linear equations to have a non-trivial solution, the determinant of its coefficient matrix must be zero.

1. Key Concept: Non-Trivial Solutions for Homogeneous Systems

A system of linear equations of the form $AX = 0$ (where $A$ is the coefficient matrix, $X$ is the column vector of variables, and $0$ is the zero vector) is called a homogeneous system.

• It always has a trivial solution, $X=0$ (i.e., $x=0, y=0, z=0$).
• For a homogeneous system to have a non-trivial solution (i.e.,