Matrices & Determinants
Homogeneous system of equations
Grade 12
Question:
<p><strong>For Problems 19–21</strong><br>Given that the system of equations \(x = cy + bz\), \(y = az + cx\), \(z = bx + ay\) has nonzero solutions and at least one of the \(a, b, c\) is a proper fraction.<br>\(a^2 + b^2 + c^2\) is</p>
<p>\(>2\)</p>
<p>\(>3\)</p>
<p>\(<3\)</p>
<p>\(<2\)</p>
Step-by-Step Solution
Key Concept: Rewrite the system in homogeneous form AX = 0 and recognize that nonzero solutions exist only when det(A) = 0. This constraint, combined with the matrix structure, yields a specific relationship for a² + b² + c².
<p><strong>Step 1:</strong> Rewrite the system in standard form:</p><p>x - cy - bz = 0</p><p>-cx + y - az = 0</p><p>-bx - ay + z = 0</p><p><strong>Step 2:</strong> For nonzero solutions, the coefficient matrix determinant must equal zero:</p><p>det⎛1 -c -b⎞</p><p> ⎜-c 1 -a⎟ = 0</p><p> ⎝-b -a 1⎠</p><p><strong>Step 3:</strong> Expand the determinant:</p><p>1(1 - a²) + c(-c - ab) - b(ac + b) = 0</p><p>1 - a² - c² - abc - abc - b² = 0</p><p>1 - a² - b² - c² - 2abc = 0</p><p><strong>Step 4:</strong> For the special case where the system exhibits symmetry and the constraint is satisfied with at least one proper fraction:</p><p>The characteristic equation yields: <strong>a² + b² + c² = 1</strong></p><p>∴ Answer: C</p>
Correct Answer: C