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 infinitely many solutions if:</p>
<p>(a) \(\lambda \neq 2, m \neq 3\)</p>
<p>(b) \(\lambda = 2, m \neq 3\)</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 infinitely many solutions when rank of coefficient matrix equals rank of augmented matrix and both are less than the number of unknowns.
<p><strong>Solution:</strong> For the system to have infinitely many solutions, the coefficient matrix and augmented matrix must have the same rank but less than the number of unknowns. This occurs when \(\lambda = 2\) and \(m\) can be any real number, making the system dependent.</p>
Correct Answer: d