What is the value of the determinant of the matrix A^{4}?
For the following two (02) items: Let A=\begin{bmatrix}\cos~\theta&\sin~\theta\\ -\sin~\theta&\cos~\theta\end{bmatrix}
- A. 0
- B. 1 ✓
- C. \cos~4\theta-\sin~4\theta
- D. \cos^{2}4\theta-\sin^{2}4\theta
Correct Answer: B. 1
Explanation
The determinant of matrix A is |A| = (\cos \theta)(\cos \theta) - (-\sin \theta)(\sin \theta) = \cos^2 \theta + \sin^2 \theta = 1. Utilizing the property of determinants |A^n| = |A|^n, we get |A^4| = 1^4 = 1.
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