Key Concept: Nature of Solutions for Homogeneous Linear Systems

A system of linear equations is called homogeneous if all the constant terms are zero. Such a system can be written in matrix form as $AX = 0$, where $A$ is the coefficient matrix, $X$ is the column vector of variables, and $0$ is the zero column vector.

For a homogeneous system $AX = 0$:

1. If the determinant of the coefficient matrix, $\Delta = \det(A)$, is non-zero ($\Delta \neq 0$), the system has only the trivial solution ($x=0, y=0, z=0$). This is a unique solution.
2. If the determinant of the coefficient matrix,