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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Matrices

NDA 2024 · Mathematics · Matrices

If A=[ array ccc 2 θ & 2 ^2 θ-1 & 0 2 θ & 2 θ θ & 0 0 & 0 & 1 array...

If A=\left[sin⁡2θ2sin⁡2θ−10cos⁡2θ2sin⁡θcos⁡θ0001\begin{array}{ccc}\sin 2 \theta & 2 \sin ^2 \theta-1 & 0 \\ \cos 2 \theta & 2 \sin \theta \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right], then which of the following statements is/are correct? 1. A−1=adj⁡A2.Aisskew−symmetricmatrix3.A−1=AT\mathrm{A}^{-1}=\operatorname{adj} \mathrm{A} 2. A is skew-symmetric matrix 3. \mathrm{A}^{-1}=\mathrm{A}^{\mathrm{T}} Select the correct answer using the code given below:

Options

  1. A1 only
  2. B1 and 2
  3. C1 and 3
  4. D2 and 3

Correct answer

C. 1 and 3

Step-by-step solution

|\mathrm{A}|=\left|sin⁡2θ−cos⁡2θ0cos⁡2θsin⁡2θ0001\begin{array}{ccc}\sin 2 \theta & -\cos 2 \theta & 0 \\ \cos 2 \theta & \sin 2 \theta & 0 \\ 0 & 0 & 1\end{array}\right|=1 We have if ∣A∣=±1|\mathrm{A}|= \pm 1 then A is orthogonal matrix. So, A−1=AT\mathrm{A}^{-1}=\mathrm{A}^{\mathrm{T}} and A−1=adj⁡A\mathrm{A}^{-1}=\operatorname{adj} \mathrm{A}.
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