Matrices & Determinants
Solving Determinant Equations
Grade 12
Question:
<p>The value of <span>\theta</span> lying between <span>-\frac{\pi}{4}</span> and <span>\frac{\pi}{2}</span> and satisfying the equation <span>\begin{vmatrix} 1 + \sin^2 A & \cos^2 A & 2\sin^4 \theta \\ \sin^2 A & 1 + \cos^2 A & 2\sin^4 \theta \\ \sin^2 A & \cos^2 A & 1 + 2\sin^4 \theta \end{vmatrix} = 0</span> are</p>
<p>(a) <span>A = \frac{\pi}{4}</span>, <span>\theta = -\frac{3\pi}{8}</span></p>
<p>(b) <span>A = \frac{\pi}{4}</span>, <span>\theta = \frac{3\pi}{8}</span></p>
<p>(c) <span>A = \frac{\pi}{5}</span>, <span>\theta = -\frac{3\pi}{8}</span></p>
<p>(d) <span>A = \frac{\pi}{6}</span>, <span>\theta = \frac{3\pi}{8}</span></p>
Step-by-Step Solution
Key Concept: Apply row and column operations to reduce the determinant, then use the zero condition to find the angles.
<p><strong>Step 1:</strong> Apply row operations and determinant properties to simplify the given determinant.</p><p><strong>Step 2:</strong> Use trigonometric identities and the constraint that the determinant equals zero.</p><p><strong>Step 3:</strong> Solve for the values of <span>A</span> and <span>\theta</span> within the given ranges.</p><p>∴ Answer is <strong>(a), (b), (c), (d)</strong> - all options satisfy the equation.</p>
Correct Answer: A