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NDA 2025
Exam
NDA
Year
2025
Subject
Mathematics
Chapter
Matrices

NDA 2025 · Mathematics · Matrices

For the following two (02) items: Let A = pmatrix θ & θ - θ & θ...

For the following two (02) items:
Let A=(cos⁡θsin⁡θ−sin⁡θcos⁡θ)A = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}.

What is the value of the determinant of the matrix A4A^4?

Options

  1. A00
  2. B11
  3. Ccos⁡4θ−sin⁡4θ\cos 4\theta - \sin 4\theta
  4. Dcos⁡24θ−sin⁡24θ\cos^2 4\theta - \sin^2 4\theta

Correct answer

B. 11

Step-by-step solution

The determinant of matrix AA is given by:

∣A∣=cos⁡θ(cos⁡θ)−sin⁡θ(−sin⁡θ)|A| = \cos\theta(\cos\theta) - \sin\theta(-\sin\theta)

∣A∣=cos⁡2θ+sin⁡2θ=1|A| = \cos^2\theta + \sin^2\theta = 1

Using the property of determinants ∣An∣=∣A∣n|A^n| = |A|^n, we have:

∣A4∣=∣A∣4=14=1|A^4| = |A|^4 = 1^4 = 1

Answer: 11
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