Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>Consider the system of equations: \(\lambda x + y + z = 1\); \(x + \lambda y + z = \lambda\); \(x + y + \lambda z = \lambda^2\).</p><p>Now, match the following lists:</p><p><strong>List I</strong></p><p><strong>a.</strong> \(\lambda = 1\)</p><p><strong>b.</strong> \(\lambda \neq 1\)</p><p><strong>c.</strong> \(\lambda \neq 1, \lambda \neq -2\)</p><p><strong>d.</strong> \(\lambda = -2\)</p><p><strong>List II</strong><br>p. unique solution<br>q. infinite solution<br>r. No solution</p><p><strong>Codes:</strong><br>(1) a-q, b-p, c-r, d-r<br>(2) a-r, b-p, c-q, d-r<br>(3) a-r, b-r, c-q, d-p<br>(4) a-q, b-p,r, c-p, d-r</p>
<p>(1) a-q, b-p, c-r, d-r</p>
<p>(2) a-r, b-p, c-q, d-r</p>
<p>(3) a-r, b-r, c-q, d-p</p>
<p>(4) a-q, b-p,r, c-p, d-r</p>
Step-by-Step Solution
Key Concept: For a system of linear equations, analyze the coefficient matrix determinant: if det(A) ≠ 0 then unique solution; if det(A) = 0, compare rank(A) with rank(A|B) to determine infinite or no solutions.
<p><strong>Step 1:</strong> Form coefficient matrix and find determinant:</p><p>A = [[λ, 1, 1], [1, λ, 1], [1, 1, λ]]</p><p>det(A) = λ³ - 3λ + 2 = (λ - 1)²(λ + 2)</p><p><strong>Step 2:</strong> Analyze each case:</p><p><strong>Case a: λ = 1</strong></p><p>All three equations become: x + y + z = 1. This represents a plane (infinite solutions). → <strong>a-q</strong></p><p><strong>Case d: λ = -2</strong></p><p>det(A) = 0. Check augmented matrix [A|B]:</p><p>Equations: -2x + y + z = 1; x - 2y + z = -2; x + y - 2z = 4</p><p>Adding first two: -x - y + 2z = -1, so x + y - 2z = 1</p><p>But third equation gives: x + y - 2z = 4. Contradiction! → <strong>d-r</strong> (No solution)</p><p><strong>Case c: λ ≠ 1, λ ≠ -2</strong></p><p>det(A) ≠ 0 → <strong>c-p</strong> (Unique solution)</p><p><strong>Case b: λ ≠ 1</strong></p><p>This includes λ = -2 (no solution) and other values (unique solution). Match is ambiguous in original framing, but if interpreted strictly: when λ ≠ 1 but could be -2, solutions vary → typically <strong>b-p,r</strong></p><p>∴ Answer: <strong>(4) a-q, b-p,r, c-p, d-r</strong></p>
Correct Answer: A