Question

If \[ q_1x + b_1y + c_1z = 0, \quad a_2x + b_2y + c_2z = 0, \quad a_3x + b_3y + c_3z = 0 \] then the given system has:

• A. one trivial and one non-trivial solution
• B. no solution
• C. one solution
• D. infinite solution

Hint:

For homogeneous systems of linear equations, check the determinant of the coefficient matrix. If it's zero, the system has infinitely many solutions.

Answer 1

The Correct Option is D

Step 1: Analyzing the system of equations.
We are given a homogeneous system of linear equations with three unknowns \( x \), \( y \), and \( z \). The system is represented as follows: \[ q_1x + b_1y + c_1z = 0, \quad a_2x + b_2y + c_2z = 0, \quad a_3x + b_3y + c_3z = 0 \] For a homogeneous system of linear equations, there will always be one trivial solution: \( x = 0 \), \( y = 0 \), and \( z = 0 \). However, the system can also have non-trivial solutions depending on the relationship between the coefficients.

Step 2: The determinant condition.
The system of equations will have non-trivial solutions if and only if the determinant of the coefficient matrix is zero. This is because a non-zero determinant implies that the system has only the trivial solution. The coefficient matrix for the system is: \[ \begin{pmatrix} q_1 & b_1 & c_1 a_2 & b_2 & c_2 a_3 & b_3 & c_3 \end{pmatrix} \]

Step 3: Determinant analysis.
If the determinant of this matrix is zero, the system will have infinite solutions. The condition for an infinite number of solutions is when the rows of the coefficient matrix are linearly dependent. This happens when the rank of the matrix is less than 3. Thus, when the determinant of the matrix is zero, the system will have infinitely many solutions.

Step 4: Conclusion.
Given the condition for non-trivial solutions, the correct answer is that the system has infinite solutions.