Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Let <em>α</em>, <em>λ</em>, <em>μ</em> ∈ <em>R</em>. Consider the system of linear equations<br>\(\alpha x + 2y = \lambda\)<br>\(3x - 2y = \mu\)<br>Which of the following statement(s) is (are) correct?<br>(1) If \(\alpha = -3\), then the system has infinitely many solutions for all values of \(\lambda\) and \(\mu\)<br>(2) If \(\alpha \neq -3\), then the system has a unique solution for all values of \(\lambda\) and \(\mu\)<br>(3) If \(\lambda + \mu = 0\), then the system has infinitely many solutions for \(\alpha = -3\)<br>(4) If \(\lambda + \mu \neq 0\), then the system has no solution for \(\alpha = -3\)</p>
<p>(1) If \(\alpha = -3\), then the system has infinitely many solutions for all values of \(\lambda\) and \(\mu\)</p>
<p>(2) If \(\alpha \neq -3\), then the system has a unique solution for all values of \(\lambda\) and \(\mu\)</p>
<p>(3) If \(\lambda + \mu = 0\), then the system has infinitely many solutions for \(\alpha = -3\)</p>
<p>(4) If \(\lambda + \mu \neq 0\), then the system has no solution for \(\alpha = -3\)</p>
Step-by-Step Solution
Key Concept: A system of 2 linear equations has a unique solution when det(A) ≠ 0, infinitely many solutions when det(A) = 0 AND the equations are proportional, and no solution when det(A) = 0 BUT equations are inconsistent. Calculate det(A) = α(-2) - 2(3) = -2α - 6 = -2(α + 3).
<p><strong>Step 1: Find the determinant of coefficient matrix</strong></p><p>A = [α, 2; 3, -2], so det(A) = -2α - 6 = -2(α + 3)</p><p><strong>Step 2: Analyze when det(A) ≠ 0</strong></p><p>If α ≠ -3, then det(A) ≠ 0, so the system has a unique solution for ANY values of λ and μ. <strong>Statement (2) is CORRECT</strong></p><p><strong>Step 3: Analyze when α = -3 (det(A) = 0)</strong></p><p>When α = -3, equations become: -3x + 2y = λ and 3x - 2y = μ</p><p>Adding them: 0 = λ + μ</p><p><strong>Step 4: Check consistency when α = -3</strong></p><p>• If λ + μ = 0: The equations are consistent (equivalent to 3x - 2y = μ). System has infinitely many solutions. <strong>Statement (3) is CORRECT</strong></p><p>• If λ + μ ≠ 0: The equations are contradictory (0 ≠ λ + μ). System has no solution. <strong>Statement (4) is CORRECT</strong></p><p><strong>Step 5: Check statement (1)</strong></p><p>When α = -3, the system does NOT have infinitely many solutions for all λ, μ (only when λ + μ = 0). <strong>Statement (1) is INCORRECT</strong></p><p>∴ Answer: (2), (3), (4)</p>
Correct Answer: (2), (3), (4)