Matrices & Determinants
Homogeneous Systems
Grade 12
Question:
<p>If \(a \geq b \geq c\) and the system of equations \(ax + by + cz = 0\), \(bx + cy + az = 0\), \(cx + ay + bz = 0\) has a non-trivial solution, then what can be said about the roots of the quadratic equation \(at^2 + bt + c = 0\)?</p>
<p>(a) Both roots are real</p>
<p>(b) Both roots are positive</p>
<p>(c) The roots are of opposite sign</p>
<p>(d) The roots are complex</p>
Step-by-Step Solution
Key Concept: Non-trivial solutions exist when the determinant equals zero, which constrains the coefficients of the quadratic.
<p>For a non-trivial solution to exist, the determinant of the coefficient matrix must be zero. This gives a constraint on $a, b, c$ that implies the discriminant of $at^2 + bt + c = 0$ is non-negative.</p>
Correct Answer: A