Matrices & Determinants
Homogeneous system of equations
Grade 12
Question:
<p>Given: <br>\(x + ky + 3z = 0,\; 3x + ky - 2z = 0,\; 2x + 4y - 3z = 0\)<br>For non-zero solutions, find the value of \(k\). Hence find \(\dfrac{xz}{y^2}\).</p>
<p>10</p>
<p>-10</p>
<p>5</p>
<p>-5</p>
Step-by-Step Solution
Key Concept: For a homogeneous system to have non-trivial solutions, the coefficient matrix determinant must equal zero. Once k is found, use the two independent equations to express x, y, z in terms of a parameter.
<p><strong>Step 1: Set up the determinant condition</strong></p><p>For non-zero solutions of the homogeneous system:<br>$$\begin{vmatrix} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 4 & -3 \end{vmatrix} = 0$$</p><p><strong>Step 2: Expand the determinant</strong></p><p>$$1(k(-3) - (-2)(4)) - k(3(-3) - (-2)(2)) + 3(3(4) - k(2)) = 0$$</p><p>$$1(-3k + 8) - k(-9 + 4) + 3(12 - 2k) = 0$$</p><p>$$-3k + 8 + 5k + 36 - 6k = 0$$</p><p>$$-4k + 44 = 0$$</p><p>$$k = 11$$</p><p><strong>Step 3: Substitute k = 11 and solve</strong></p><p>The system becomes:<br>$$x + 11y + 3z = 0 \quad (1)$$<br>$$3x + 11y - 2z = 0 \quad (2)$$</p><p>From (1) − (2): $$-2x + 5z = 0 \Rightarrow x = \frac{5z}{2}$$</p><p>Substitute into (1): $$\frac{5z}{2} + 11y + 3z = 0$$</p><p>$$11y = -\frac{11z}{2} \Rightarrow y = -\frac{z}{2}$$</p><p><strong>Step 4: Calculate the ratio</strong></p><p>$$\frac{xz}{y^2} = \frac{\frac{5z}{2} \cdot z}{\left(-\frac{z}{2}\right)^2} = \frac{\frac{5z^2}{2}}{\frac{z^2}{4}} = \frac{5z^2}{2} \cdot \frac{4}{z^2} = 10$$</p><p>∴ Answer: <strong>k = 11 and xz/y² = 10</strong></p>
Correct Answer: A