Matrices & Determinants
Homogeneous Linear Equations
Grade None
Question:
<p>If the system of linear equations</p><p>\(x + 2ay + az = 0\)</p><p>\(x + 3by + bz = 0\)</p><p>\(x + 4cy + cz = 0\)</p><p>has a non-zero solution, then \(a\), \(b\), \(c\) are in:</p>
<p>(a) A.P.</p>
<p>(b) G.P.</p>
<p>(c) H.P.</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: For non-trivial solutions in homogeneous systems, the coefficient matrix determinant must equal zero, which constrains the parameters.
<p><strong>Solution:</strong> For a homogeneous system to have non-trivial solutions, the determinant of the coefficient matrix must be zero:</p><p>$$\begin{vmatrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \end{vmatrix} = 0$$</p><p>Expanding and simplifying using row operations leads to the condition that $a$, $b$, $c$ satisfy the relationship for Harmonic Progression (H.P.).</p>
Correct Answer: C