Matrices & Determinants
System of Linear Equations
Grade 12
Question:
<p>The values of \(k \in R\) for which the system of equations \(x + ky + 3z = 0\), \(kx + 2y + 2z = 0\), \(2x + 3y + 4z = 0\) has nontrivial solution are</p>
<p>2</p>
<p>5/2</p>
<p>3</p>
<p>5/4</p>
Step-by-Step Solution
Key Concept: A homogeneous system has nontrivial solutions if and only if the determinant of the coefficient matrix equals zero. Calculate det(A) and set it equal to zero to find the values of k.
<p><strong>Step 1:</strong> For a homogeneous system Ax = 0 to have nontrivial solutions, we need det(A) = 0.</p><p><strong>Step 2:</strong> Set up the coefficient matrix and calculate its determinant:</p><p>$$\begin{vmatrix} 1 & k & 3 \\ k & 2 & 2 \\ 2 & 3 & 4 \end{vmatrix} = 0$$</p><p><strong>Step 3:</strong> Expand along the first row:</p><p>$$1(2 \cdot 4 - 2 \cdot 3) - k(k \cdot 4 - 2 \cdot 2) + 3(k \cdot 3 - 2 \cdot 2) = 0$$</p><p>$$1(8 - 6) - k(4k - 4) + 3(3k - 4) = 0$$</p><p>$$2 - 4k^2 + 4k + 9k - 12 = 0$$</p><p>$$-4k^2 + 13k - 10 = 0$$</p><p>$$4k^2 - 13k + 10 = 0$$</p><p><strong>Step 4:</strong> Factor or use quadratic formula:</p><p>$$(4k - 5)(k - 2) = 0$$</p><p>$$k = \frac{5}{4} \text{ or } k = 2$$</p><p>∴ Answer: A</p>
Correct Answer: A