Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Find <math>k</math> such that the following system of equations has infinitely many solutions:</p><p><math>3x - 2y - kz = 0</math></p><p><math>2x - 4y - 2z = 0</math></p><p><math>x + 2y - z = 0</math></p>
Step-by-Step Solution
Key Concept: When the determinant is zero and non-trivial solutions exist, the value of parameter k ensures consistency of the system.
<p><strong>For infinitely many solutions:</strong> <math>D = 0</math></p><p><math>D = \begin{vmatrix} 3 & -2 & -k \\ 2 & -4 & -2 \\ 1 & 2 & -1 \end{vmatrix} = 0</math></p><p><strong>Expanding:</strong></p><p><math>3(4 + 4) + 2(-2 + 2) - k(4 + 4) = 0</math></p><p><math>3(8) - 8k = 0</math></p><p><math>24 = 8k</math></p><p><math>k = 3</math></p><p><strong>Verification:</strong> With <math>k = 3</math>, the equations reduce to:</p><p><math>x + y + 3z = 0</math> and <math>2y + 6z = 0</math></p><p>This gives <math>y = -3z</math> and <math>x = 0</math>.</p>
Correct Answer: 3