Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Find the value of <math>\lambda</math> such that the system of equations has infinitely many (non-trivial) solutions:</p><p><math>3x - 2y + z = 0</math></p><p><math>\lambda x - 14y + 15z = 0</math></p><p><math>x + 2y - 3z = 0</math></p>
Step-by-Step Solution
Key Concept: For a homogeneous system to have infinitely many non-trivial solutions, the coefficient matrix determinant must equal zero.
<p><strong>For infinitely many non-trivial solutions:</strong> The determinant <math>D = 0</math></p><p><math>D = \begin{vmatrix} 3 & -2 & 1 \\ \lambda & -14 & 15 \\ 1 & 2 & -3 \end{vmatrix} = 0</math></p><p><strong>Expanding along first row:</strong></p><p><math>3(42 - 30) - \lambda(6 - 2) + 1(-30 + 14) = 0</math></p><p><math>3(12) - 4\lambda - 16 = 0</math></p><p><math>36 - 4\lambda - 16 = 0</math></p><p><math>4\lambda = 20</math></p><p><math>\lambda = 5</math></p>
Correct Answer: 5