Question

Consider the system of equations a₁x + b₁y + c₁z = 0, a₂x + b₂y + c₂z = 0, a₃x + b₃y + c₃z = 0 if

$$\begin{vmatrix} a_1 & b_1 & c_1\\ a_2 & b_2 & c_2\\ a_3 & b_3 & c_3 \end{vmatrix} = 0,$$

then the system has

• A. more than two solutions
• B. only non trivial solutions
• C. no solution
• D. only trivial solution (0, 0,0)

Answer 1

Answer: (A)

more than two solutions

Answer 2

Key Observation: This is a homogeneous system $\mathbf{A}\mathbf{x} = \mathbf{0}$.
Determinant Zero: $\Delta = \det(\mathbf{A}) = 0$.

• For a homogeneous system, if $\Delta = 0$ then there are infinitely many (i.e.\ more than two) solutions.
• We have $\Delta_x=\Delta_y=\Delta_z=0$ automatically, confirming non-uniqueness.
  Conclusion: The system has more than two solutions.