Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>If the trivial solution is the only solution of the system of equations<br/>\(x + ky + z = 0\)<br/>\(kx + 3y + kz = 0\)<br/>\(3x + y + z = 0\)<br/>Then, the set of values of \(k\) is</p>
<p>(a) \(\{2, -3\}\)</p>
<p>(b) \(\mathbb{R} \setminus \{2, -3\}\)</p>
<p>(c) \(\mathbb{R} \setminus \{2\}\)</p>
<p>(d) \(\mathbb{R} \setminus \{-3\}\)</p>
Step-by-Step Solution
Key Concept: A homogeneous system has only the trivial solution when the determinant of the coefficient matrix is non-zero.
<p><strong>Solution:</strong> For the trivial solution to be the only solution, the determinant of the coefficient matrix must be non-zero. Calculate the determinant and find when it is not equal to zero.</p>
Correct Answer: B