Matrices & Determinants
Determinant with Trigonometric Terms
Grade 12
Question:
<p>If \(A\), \(B\), \(C\) are the angles of triangle \(ABC\), then the minimum value of <span class="math">\begin{vmatrix} -2 & \cos C & \cos B \\ \cos C & -1 & \cos A \\ \cos B & \cos A & -1 \end{vmatrix}</span> is equal to:</p>
<p>(a) 0</p>
<p>(b) -1</p>
<p>(c) 1</p>
<p>(d) -2</p>
Step-by-Step Solution
Key Concept: Apply triangle angle constraints and use determinant properties for symmetric matrices with trigonometric entries.
<p><strong>Solution:</strong> Use the constraint that $A + B + C = \pi$ in a triangle. Evaluate the determinant using properties of symmetric matrices and trigonometric identities. The minimum value occurs at specific angle configurations and equals -1.</p>
Correct Answer: B