Matrices & Determinants
Determinants and Non-trivial Solutions
Grade 12
Question:
<p>If the system of linear equations \[x + ky + 3z = 0\]\[3x + ky - 2z = 0\]\[2x + 4y - 3z = 0\] has a non-zero solution \((x, y, z)\), then \(\frac{xz}{y^2}\) is equal to</p>
<p>(a) -10</p>
<p>(b) 10</p>
<p>(c) -30</p>
<p>(d) 30</p>
Step-by-Step Solution
Key Concept: For a homogeneous system to have non-zero solutions, the determinant of the coefficient matrix must equal zero. Use this condition to find the parameter and then compute the required ratio.
<p><strong>Analysis:</strong> For a homogeneous system to have non-zero solution, the determinant of coefficients must be zero. Set up and solve $\det\begin{pmatrix}1 & k & 3\\3 & k & -2\\2 & 4 & -3\end{pmatrix} = 0$ to find the value(s) of <em>k</em>. Then substitute back and find the relationship between <em>x</em>, <em>y</em>, <em>z</em> to calculate $\frac{xz}{y^2}$.</p>
Correct Answer: C