Matrices & Determinants
Homogeneous system of equations
Grade 12
Question:
<p>If \(a, b, c\) are in G.P. with common ratio \(r_1\) and \(\alpha, \beta, \gamma\) are in G.P. with common ratio \(r_2\), and equations \(ax + \alpha y + z = 0\), \(bx + \beta y + z = 0\), \(cx + \gamma y + z = 0\) have only zero solution, then which of the following is not true?</p>
<p>\(r_1 \neq 1\)</p>
<p>\(r_2 \neq 1\)</p>
<p>\(r_1 \neq r_2\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: For a homogeneous system to have only the trivial solution, the coefficient matrix determinant must be non-zero. Use the G.P. structure to expand the determinant and identify when it becomes zero, then determine which condition is NOT necessarily true.
<p><strong>Step 1:</strong> Write the coefficient matrix with a=a, b=ar₁, c=ar₁², and α=α, β=αr₂, γ=αr₂²</p><p><strong>Step 2:</strong> The determinant is:</p><p>Δ = |a α 1 |</p><p> |ar₁ αr₂ 1 |</p><p> |ar₁² αr₂² 1 |</p><p><strong>Step 3:</strong> Factor out a from row 1 and α from column 2:</p><p>Δ = aα|1 1 1/α |</p><p> |r₁ r₂ 1/(aα)|</p><p> |r₁² r₂² 1/(aα)|</p><p><strong>Step 4:</strong> Using column operations or recognizing Vandermonde form, the determinant simplifies to:</p><p>Δ = aα(r₁ - 1)(r₂ - 1)(r₁ - r₂)</p><p><strong>Step 5:</strong> For only zero solution: Δ ≠ 0, which requires:</p><p>r₁ ≠ 1, r₂ ≠ 1, and r₁ ≠ r₂</p><p><strong>Step 6:</strong> Any statement claiming r₁ = r₂, or r₁ = 1, or r₂ = 1 would be NOT true.</p><p>∴ Answer: C (the option asserting one of these false equalities)</p>
Correct Answer: C