Matrices & Determinants
Linear System Solutions
Grade 12
Question:
<p>Find the values of \(\lambda\) and \(b\) for which the system of equations \(x + y + z = 3\), \(x + 3y + 2z = 6\), \(x + \lambda y + 3z = b\) has:</p>
<p>(a) A unique solution, if \(\lambda \neq 5\), \(b \in \mathbb{R}\)</p>
<p>(b) No solution, if \(\lambda = 5\), \(b \neq 9\)</p>
<p>(c) Infinitely many solutions, if \(\lambda = 5\), \(b = 9\)</p>
<p>(d) All of the above</p>
Step-by-Step Solution
Key Concept: Use the determinant to classify solutions: nonzero determinant ⟹ unique solution; zero determinant requires rank analysis for infinitely many or no solutions.
<p><strong>Step 1:</strong> Compute the determinant of the coefficient matrix. For $\lambda \neq 5$, the determinant is nonzero, giving a unique solution.<br/><strong>Step 2:</strong> For $\lambda = 5$, the determinant is zero. Verify using Rouché–Capelli theorem: inconsistent when $b \neq 9$, infinitely many solutions when $b = 9$.</p>
Correct Answer: D