Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Consider the system of equations<br/>\[2x + \lambda y + 6z = 8\]
\[x + 2y + mz = 5\]
\[x + y + 3z = 4\]<br/>The system of equations has exactly one solution if:</p>
<p>(a) \(\lambda \neq 2, m \neq 3\)</p>
<p>(b) \(\lambda = 2, m = 3\)</p>
<p>(c) \(\lambda \neq 2, m = 3\)</p>
<p>(d) \(\lambda = 2, m \in \mathbb{R}\)</p>
Step-by-Step Solution
Key Concept: A system has exactly one solution when the coefficient matrix has full rank (non-zero determinant).
<p><strong>Solution:</strong> For the system to have exactly one solution, the coefficient matrix must be non-singular, which requires \(\lambda \neq 2\) and \(m \neq 3\), ensuring the determinant is non-zero.</p>
Correct Answer: a