This problem involves a system of linear homogeneous equations. For such a system to have a non-zero (or non-trivial) solution, the determinant of the coefficient matrix must be zero. We will set up the determinant, simplify it using row and column operations, and then expand it to find the relation between $a, b, c$.

1. Key Concept: Non-trivial Solution for Homogeneous System

A system of linear homogeneous equations, represented as $AX = 0$, where $A$ is the coefficient matrix, $X$ is the column vector of variables, and $0$ is the zero vector, has a non-zero (non-trivial) solution if and only if the determinant of the coefficient matrix, $\det(A)$, is equal to zero.