Matrices & Determinants
Special Matrices
Grade 12
Question:
<p>Let \(\theta = \frac{\pi}{5}\) and \(A = \begin{pmatrix}\cos\theta & \sin\theta\\-\sin\theta & \cos\theta\end{pmatrix}\). If \(B = A + A^4\), then \(\det(B)\) is</p>
<p>(a) is one</p>
<p>(b) lies in (2, 3)</p>
<p>(c) is zero</p>
<p>(d) lies in (1, 2)</p>
Step-by-Step Solution
Key Concept: For rotation matrices, $\det(A) = 1$. The determinant of the sum depends on the specific angles involved; here the geometry of angles $\frac{\pi}{5}$ and $\frac{4\pi}{5}$ makes B singular.
<p><strong>Analysis:</strong> The matrix $A$ is a rotation matrix with determinant 1. The matrix $A^4$ is also a rotation matrix by angle $4\theta = \frac{4\pi}{5}$. The matrix $B = A + A^4$ represents the sum of two rotation matrices. With $\theta = \frac{\pi}{5}$, the angles 1 and 4 are related such that $\det(B) = 0$.</p>
Correct Answer: C