Matrices & Determinants
Homogeneous system of linear equations
Grade 12
Question:
<p>The greatest value of <math>c \in \mathbb{R}</math> for which the system of linear equations <math>x - cy - cz = 0</math>, <math>cx - y + cz = 0</math>, <math>cx + cy - z = 0</math> has a non-trivial solution, is</p>
<p>(a) -1</p>
<p>(b) <math>\frac{1}{2}</math></p>
<p>(c) 2</p>
<p>(d) 0</p>
Step-by-Step Solution
Key Concept: For a homogeneous system to have non-trivial solutions, the determinant of coefficients must equal zero. Find the value of parameter c that makes D = 0.
<p><strong>Step 1:</strong> A homogeneous system of linear equations has non-trivial solutions when <math>D = 0</math>.</p><p><strong>Step 2:</strong> For the given system, set up the determinant:</p><p><math>D = \begin{vmatrix} 1 & -c & -c \\ c & -1 & c \\ c & c & -1 \end{vmatrix} = 0</math></p><p><strong>Step 3:</strong> Expand and simplify to find the value(s) of <math>c</math> for which <math>D = 0</math>.</p><p><strong>Step 4:</strong> The greatest value of <math>c</math> satisfying this condition is <math>\frac{1}{2}</math>.</p><p>∴ Answer is (b).</p>
Correct Answer: B